My reading of this Q&A is that elves recover spell slots after a 4-hour long trance.
However, if that is the case, then what good is Arcane Recovery to an Elf Wizard?
There really is only one benefit of the Arcane Recovery feature over taking a Long Rest (4 Hour trance for Elves) and that is time. A short rest as defined by the PHB is:
A short rest is a period of downtime, at least 1 hour long, during which a character does nothing more strenuous than eating, drinking, reading and tending to wounds.
If we then look at Arcane Recovery and what it grants us:
You have learned to regain some of your magical energy by studying your spellbook. Once per day when you finish a short rest, you can choose expended spell slots to recover. The spells slots can have a combined level that is equal to or less than half your wizard level.
So the benefits of Arcane Recovery are the same for an Elf wizard as they are for any other race of wizard. The benefit of the trance is allowing a long rest in 4 hours, it does not modify the utility of a short rest which only takes 1 hour.
As pointed out by @Joel Harmon and Dan B. The rules on Long Rests apply to the Elf's trace feature and contain the following:
A character can't benefit from more than one long rest in a 24-hour period.
If you have 4 hours to rest and have used neither your long rest or Arcane Recovery within the last 24 hours then yes, the Arcane Recovery feature won't help you that much since you can regain all of your slots with a trance instead of just some with a short rest.
If you don't have 4 hours, have already had a long rest, or can't convince the rest of your party to wait that long then Arcane Recovery will help you just like it will help any other wizard.
Since both spell recovery methods are limited to once per 24 hours it makes sense to use both of them to regain the maximum number of spell slots you can.
Arcane Recovery is just as useful for an elf wizard as it would be for any other wizard.
If I'm understanding your question correctly, it sounds like you're asking this:
"My elf wizard can regain all his spell slots by taking a four-hour long rest. So why should he bother regaining just a few spell slots in one hour with Arcane Recovery, when he can ask the party to rest just three hours longer and get them all back?"
The point you're missing is this text from the Long Rest rules (linked in your question):
You cannot benefit from more than one long rest in a 24-hour period.
Your party will be long-resting once per day anyway, so you can't get any more long rests than that.
Your "finish a long rest in four hours" ability means you can spend more time staying on watch during the night while everyone else sleeps, but unfortunately that's the only benefit you're likely to get from it.
The elf wizard is (typically) one of many party members; each of those party members might recover certain resources on a short rest or a long rest. After a fight or two, it should be easy to convince the party to take a short rest.
Some classes, like fighters, monks, and warlocks, recover all or most of their resources on a short rest. Most classes at least recover something, as with the wizard's Arcane Recovery. Everyone can at least roll Hit Dice to recover HP. It's a lot easier to say "let's all take a short rest" than "you all take a short rest followed by 3 hours of nothing, while I recover spell slots".
The difference between 4 hours and 1 hour may determine if the evil necromancer completes her ritual by midnight. You may be able to squeeze in a short rest when you don't have time to wait for 4 hours. Well-designed adventures may account for this by providing just enough time for a fast party to take one or two short rests, perhaps with consequences for doing so.
Bards, sorcerers, and wizards have access to the spell catnap (XGtE, p. 151), which grants the benefits of a short rest to three creatures over the course of 10 minutes.
Additionally, the Dungeon Master's Guide presents the Epic Heroism rest variant as a play option, in which a short rest only lasts 5 minutes (DMG, p. 267).
As Joel Harmon pointed out in the comments of linksassin's answer, you may only benefit from one long rest in a 24-hour period (PHB, p. 186). There is no such restriction on short rests; however, linksassin rightly points out in the comments that you may only use Arcane Recovery once per day (which may or may not be the same as once per long rest).
Additionally, you need not sleep (or trance) during a short rest; you can remain fully alert and aware. Trancing elves are "semiconscious" during their long rest, which a DM may rule imposes disadvantage on Perception checks.