I normally avoid recommending generic, "universal" systems for questions like this. It's too easy to think of one's own favourite generic system and think that it can do whatever is being asked about. I hesitate, but not all generic systems are made equal – each has different fiddly bits that it brings to the experience of play, and in this case, Savage Worlds has a few core features that line up very nicely with your requirements.
Savage Worlds is not everyone's cup of tea though – people seem to either love it or hate it, usually based on whether the other bits of the system line up with one's own taste or not. I can't guess whether you'll like it or not, so I can only recommend that you try it – and fortunately, there is a free "Test Drive" version of the SW rules which, paired with a fan conversion of Starship Troopers gear, troopers, and bugs, provides you with enough to give it a spin some evening to see if it lights your fire or if it rubs you the wrong way. Free is pretty good for an evening or more's entertainment, and the rules themselves are an affordable $10 (in print or PDF) for when you find that the Test Drive doesn't give you the guidance or options you want.
But let me tell you about these not-so-generic parts of the system that are suited to what you like in Starship Troopers.
Skirmish combat
SW evolved out of skirmish-level miniature rules for an entirely different RPG, eventually becoming its own roleplaying game with a tactical skirmish combat engine at its core. Though it's built around individual characters, the details of the combat system were all chosen to make handling large numbers of allies and enemies very quick, so that turns go around the table at a brisk pace and the action keeps flowing.
Now, the exact details of how it does this is also exactly where a lot of people either love or hate the system. I can only say, "try it," and then you'll see if you like it or not.
Specifically, the details of the combat system relevant to your first requirements bullet:
It uses miniatures and measures distance/movement by inches, so you can use a grid or you can use a tape measure. On the plus side, the combat system isn't tightly reliant on miniatures in the way that (for example) D&D 4e is, so you can just as easily run combats narratively. Using miniatures just adds precision to the positional tactics of the game, rather than being required to play tactically.
Combat rolls are fairly quick. There is more than one roll, but it trades a couple more rolls for eliminating the usual bookkeeping involved in combats. There are no hit-points to track, only one status (and it doesn't cause modifiers), and it expects most opponents to be Extras, using the "mook" rules so that they're either up, down, or out of play. Important individuals have multiple wounds (always exactly 3). If you've played very many miniature skirmish games, this should look familiar – regular soldiers are either alive or not, while important figures can take a few hits before going down.
You can have lots of figures in play at once and the combat system slows down logarithmically (i.e., the rate at which the game slows down is slight as you add more combatants) instead of exponentially like in most systems. Part of this is the mook rules, but part of this is also that you can only do so much in one turn. Multiple combatants is handled well, and ganging up on an single opponent gives significant advantages. Being surrounded by bugs is not a good idea, and some of the tactical considerations are going to be at the level of battlefield control, to avoid having the better-armed troopers getting separated, surrounded, and physically dragged down.
The system is tactical both in terms of positioning, but also in terms of how you use your actions. Rather than getting a complete "what are you doing" each round, to be effective you have to chain together rounds for what you're trying to accomplish. You can just stand toe-to-toe and fight, but this is the way to lose a fight in SW; a tactical fighter will set up their position, split up their enemy with suppressing fire, charge into melee, duck inside their reach, and then fire into the bug's soft underbelly and take it out in one go. Similarly, a big bug might charge while roaring (which can unnerve the troopers), slam the ground to knock them down, and sweep to attack the softened troopers; there are many possible tactics.
There are built-in ways to gather together a number of bonuses (yourself and from allies) that you can then use all at once for a single devastating attack, and this is the expected way to use the combat system. A naïve player might try to toe-to-toe and just attack every round hoping for a lucky hit, but this is putting oneself at the mercy of enemies who fight smarter. Since there are no hit points to whittle away, it's all about teamwork and smart manœuvering so that you can hit with a lot of bonuses all at once and take them out with a single attack roll. You can also do multiple things in one round at the expense of penalties the more you try to do, and this is sometimes a great idea if you've previously set yourself up with a lot of bonuses to offset them. There are a lot of natural trade-offs that you can make during combat, and how effective a trade-off decision will be is entirely situational; this gives you a lot of possible action-tactics from a relatively small number of interacting options, and the difference between good tactics and poor tactics is a matter of properly reading the situation (just like real combat) rather than finding a "kill combo" that you can use in every fight.
Setting
The specific sci-fi elements of Heinlein's Starship Troopers are not readily available. (Well, except they are, because someone has already done a bug and gear conversion for the movie and you can use many of those stats unaltered.) But, the system is extremely quick to stat up gear and opposition, which means that you can have a working conversion of the setting to SW terms in less than an hour's work (maybe much less), given familiarity with the system. Opposition is very easy to stat up – fast enough that you can do it on-the-fly while improvising an encounter. A Savage Worlds GM with only a little experience can stat up even an unusual creature in about two minutes, which is little enough that you can do it in between narrations or while the players discuss their plan of attack.
Gear is simple and has few moving parts. Again, if you're familiar with sci-fi wargames, you'll find Savage Worlds gear to be familiar. There are few stats and their effects are sufficiently large-grained that you can eyeball changes easily. Special effects are as easy as saying "it does this" or "it gives you this Edge / the equivalent of this Power". Being a toolkit system with effect-based abilities, you can as easily use the (e.g.) Bolt Power's rules to stat up a magic missile spell or a plasma carbine's primary firing mode, and you can do it on the order of a minute. A multitude of weapons is easy to create and the system is made for just eyeballing the numbers to fit the setting you're using; just think of the effect you want it to have, and give it those abilities and stats to support the concept. You can use the Powers and Edges in the book to model weapon functions, or you can just define them ad hoc by your own judgement. It's rather hard to break the system even when modifying its rules, let alone when just statting up new stuff.
Another feature/flaw of Savage Worlds that works to your advantage here is that the genre and tone of the game are not dictated by the system, and is entirely on your shoulders to convey. It means that it won't give you any help creating the feel of Starship Troopers, but it also won't get in the way of you just running it according to how you know Starship Troopers feels. This is another reason why the fan conversion for the movie is useful to you at all: the feel and tone isn't built into the gear and enemy stats, so the feel of the movie isn't glued to the fan conversion. (The jump jets are missing though: just say that the battle dress also gives you the Fly Power with a fixed duration of 1 or 2 rounds and a fixed movement limit per jump and you're good to go!)
Non-combat
There are social skills and a social "combat" mechanic that you can use if you like. There's enough bits in the system that you can use it outside of combat for things like facing the wrath of your commanding officer while you disagree with them, mechanically handling the arguments and evidence during a court martial, or debating politics and trying to convert the people overhearing you to your point of view. And yup, there's a Repair skill.
Niche "protection"
The game is not class based, yet it still offers significant niche-adoption. I won't say niche protection because nothing is stopping the, say, medic from putting in the time and effort to become a good sniper, but that choice means they're making a trade-off: they're not focusing on doing their medicking better. The rules support niche-adoption because becoming good at something requires investing in that something with supporting Edges; it's not enough to up your stats or skills (actually, it's often worse to up a skill or stat when you could instead get a +2 from an Edge), so once a character has created a niche, it behooves them to continue to strengthen their abilities in that niche and to choose complementary support Edges and skill improvements. Often you can move laterally – a good sniper can also branch out to become an infiltrator, for example, because not being seen is an overlapping competency of those niches – but it's very hard to satisfyingly move to a completely different niche that has no overlapping competencies. You just end up having a bunch of skills and Edges that you can't really use in your new niche, so why do that?
Besides, it's much more fun to simply run multiple characters. NPCs under a PC's command are traditionally controlled by that PC's player, so if a player with a sniper PC wants to roleplay some medic situations, then it's much better to give them a medic NPC to command (maybe as part of a squad of infiltrators) than to twist their perfectly-good sniper PC into an ineffective sniper-medic.
Miscellaneous features
Other selling points of Savage Worlds that aren't specific to your requirements, but are helpful for what you're aiming to do:
Characters are quick to make. They will be a bit slow at first because an unfamiliar system always slows down character creation, but it will still be fairly fast. A group with two or three characters under their belts can create a party of characters in under an hour at worst, and closer to 20 minutes or less. The Starship Troopers fan conversion gives you a template of a mobile infantry member and a few customisation steps, so that should give you a party of distinct characters in much less time even with no experience with the system.
The power "curve" of the system is very flat. There is a difference between Novice and Legendary characters, but it's not exponential like in some systems. Development tends to broaden characters a lot, while raising their skills only a bit. A character with a lot of experience tends to be more reliable at doing the same things they always could instead of being orders of magnitude more powerful in those abilities. This means that the very same bugs that were a threat when they were green troops will still be an interesting threat when they are elites, and the nastiest bugs are possible to face (carefully) right from the get-go given good planning, support, and cooperative tactics.
The system is very easy to tweak, and robust in the face of house rules. Changing one or many parts of how it works doesn't cause a domino effect of unexpected breakage in other parts of the system. You can tune the various subsystems pretty much independently from each other, and they'll keep on humming along.
It's almost impossible to game the system to create "broken" characters. I know, that's hard to believe. But people have tried, and what happens every time is that what they end up with is a character that only seems broken because they don't understand SW well, but are actually broken in the "does not function well" sense. Part of this is because the game has no "dump stats" when it comes to combat – dumping Smarts or Spirit (the only "social" stats the game has) in order to pump Agility (for hitting ability), Strength (for damage output), or Vigor (for damage resistance) results in characters that are easy to fatally trick during combat, or who crumble and maybe rout as soon as the enemy gets a hit on them. Trying to game the system to make a "broken" character either makes a very fragile, unreliable combatant; or actually makes a solid, non-broken character.
Extras aren't boring. They have all the tactical options and abilities of regular characters, and they can have good stats, loads of Edges, and high-quality gear as much as any Legendary PC – they just don't have more than one wound or get the roll-smoothing benefit of the "Wild Die". Don't be afraid to make even your really nasty bugs Extras – they'll be a terror on the battlefield, and your players will cheer when they go down. Customarily only named characters aren't Extras, so even though your PCs' sergeant will be a regular, named character (the game calls them "Wild Cards", but eh), a different random sergeant with the same stats can be an Extra on the battlefield, to make your bookkeeping easier and combat go faster. Ditto your bugs – if there is a legendary bug that has never been killed, but everyone tells stories of it and it can be recognised by its scars, that bug is special, and gets the "Wild Die" and 3 wounds. Meanwhile, an identical bug of the same type can be an Extra just fine. They're still nasty and have the same stats, but facing a nameless one is just not the same degree of epic as facing Old Half-Jaw.
TL;DR
Savage Worlds is already a hybrid miniatures skirmish game / RPG. It has everything you'd expect from a skirmish-level wargame, but it's seamlessly integrated with roleplaying-game elements and gives you the system-level flexibility to create units and characters at any degree of detail you need. It's very fast to create material for, leaving you more brain left over for designing scenarios, crafting NPC personalities, devising interesting battle plans, and running political undercurrents.
