Summary
As a sort of in-between weapon, a thrown weapon has several advantages and disadvantages, but among them are the unique constraint on the number of attacks one can make and the unique feature of leaving the implementing hand unburdened. There is one character class in particular whose damage throughput is balanced around making relatively few attacks, but still with weapons, and that is the Rogue with its Sneak Attack ability. Further, as thrown weapons can double as melee, require only a single hand, and leave that hand unburdened, thrown weapons give ranged Rogues several opportunities for a second delivery of Sneak Attack damage via an Opportunity Attack if they are built to allow for such complicated tactics.
To Start Off...
I'll begin by saying that I've always thought of a ranged attack with a thrown weapon (hereafter called a "thrown weapon attack" with no mechanical import) as a peripheral option for melee characters who find themselves without a target in melee range (like a monk who draws and throws a dagger to keep their attack count up) or for any character looking to add some "cool" and efficiency to a weapon swap (like a barbarian who throws his two hand axes before going in with his greataxe the next round) - not at all as something to center a build around. After all, why would someone looking to focus on ranged attacks not use a weapon designed specifically for that rather than adapt one made for melee?
Granted, all the same could be said of unarmed strikes, and yet there are still those one or two ways to make it your character's thing. So, let's see if there's something for throwing.
My Thoughts on Thrown Weapon Advantages
One advantage to melee thrown weapons is the ability to make a ranged attack with Strength rather than Dexterity. Dexterity is normally considered the more efficient stat, given its contribution to ubiquitous AC, but this is not necessarily the case for characters who have benefits to strength, who are built to emphasize strength-based tactics, or for whom the efficiency of dexterity is diminished (for instance, a barbarian, a grappler, or character wearing heavy armor). Any such character might prefer a thrown weapon to a ranged weapon for their ranged attacks, but of course we're left with the question of why they might be making ranged attacks instead of melee attacks in the first place.
Another advantage of melee thrown weapons is their attack flexibility. To flip the "peripheral" benefit mentioned earlier, if the bias moves from melee to ranged attacks as the norm of a build, a thrown weapon has some nice utility as a melee weapon in situations where it might be less advantageous to make a ranged attack. For instance, a normally ranged character might wish they had specialized in daggers rather than hand crossbows when getting mobbed by a hostile crowd and facing all their attacks having disadvantage. Of course, they could always drop-and-draw, but then again they might want that crossbow back next round... and after all that we might be left wondering how often these kind of situations actually come up.
I think you're right about the major limitation on thrown weapons being their need to be drawn. For ranged weapons, this is the hidden benefit of the ammunition rule: while you have to draw an arrow, you also get to draw one. That thrown weapons cannot live up to a possible attack count is why strength-based characters prefer to make melee attacks and why ranged characters stick with ranged weapons. However, even within this limitation, thrown weapons have their own hidden benefit: they can be drawn and thrown using a single hand, and they leave that hand unoccupied after the attack. Per this, a thrown weapon attack could be worked in while holding a two-handed weapon, before or after a grapple attempt, or as a way to leave a hand open for spellcasting. However, the caster would have to want to throw the weapon more than to cast a spell, and etc. While reasons could be thought of for each case, they would hardly make the character centered on throwing.
What I've Come Up With
To my mind, the best way to play to the advantages of thrown weapons is to also play into mechanics that maximize the damage done with few attacks. Of such "once per turn" -style buffs, the most obvious one is Sneak Attack. Playing a Rogue at range really gets the most out of their Uncanny Dodge and Evasion abilities (as well as getting more DM-approved Hide opportunities), and Sneak Attack is basically already designed to compensate for the Rogue's lack of Extra Attack, even exceeding it if not also balanced against Sneak Attack's weapon limitations and situational requirements.
To play into Sneak Attack with a thrown weapon requires that one use a ranged thrown weapon or a finesse melee thrown weapon, since making a ranged attack with a non-finesse melee thrown weapon is not sufficient to trigger Sneak Attack. There are two ranged thrown weapons, the dart and the net, and only the dart does damage (and here I'll ignore the issue of whether a net can benefit from Sneak Attack). There is also the trusty dagger for a melee finesse thrown weapon, and while it is the only one which could be used in a strength build, it makes more sense stick with a Rogue's traditional Dex here anyway.
What I envision is a rogue character that draws and throws a dagger or dart each round, likely trying for Sneak Attack -eligible targets (and likely an Assassin-archetype for its Sneak Attack -enabling abilities). While it could hold another light weapon and two-weapon fight, or even regularly draw and throw another dagger if it took the Dual Wielder feat, my character would instead seek to play once more into Sneak Attack by exploiting a character's ability to use Sneak Attack again in a round via their Opportunity Attack. Of course, a character cannot make an opportunity attack at range, so I'll need a melee weapon. Luckily, since drawing and throwing a thrown weapon requires only one hand, my character can be holding pretty much anything else in the other (even a two-handed weapon) for use during that round's reaction. I'll select from finesse weapons for Sneak Attack, and then I'll choose either a light weapon to keep open the possibility of Two-Weapon Fighting (if I don't take Dual Wielder), a Rapier for its damage profile, or a Whip (with a dip or a feat for proficiency) for its dynamic threat-range potential.
The idea of this character is that they desire to make ranged attacks while leaving themselves in the best "hand disposition" for opportunity attacks; their ranged nature gives them the flexibility to use their position to increase the likelihood of opportunity attacks, and thrown weapons free up their hands for melee weapons with which to make those opportunity attacks. They can threaten a hallway or other choke-point, or tie up an enemy that would prefer to move away, or maneuver behind a peripheral target that's about to charge in, all while continuing to make ranged attacks against their own preferred target. Should they become "tied up," they can seamlessly transition into melee combat. The major limitation to this build is the common 20 ft. normal range increment on available thrown weapons, which will have to be kept carefully in mind.
Personally, I prefer the thrown dagger/whip setup, since I prefer to save my bonus actions for Cunning Action (again, playing to the Rogue's flexibility), and I like to use the Whip's reach to both threaten and make ranged attacks without disadvantage. Sometimes, just holding a dagger gives me another threat-range trigger layer (still only one actual opportunity attack per round, of course).
Conclusion
After training your DM with a couple of examples of this build's combat potential, it may be that no other enemy ever leaves your reach again, so you'll just have to measure its benefit by estimating the unknown alternative actions enemies might have taken were you not so threatening. Sitting at the intersection of action efficiency, feature efficiency, tactical positioning, and situational target selection, this build rewards a Rogue player's cunning in combat, and it wouldn't be possible without the unique benefits of Thrown Weapons.
