Now that the preliminaries are out of the way, let's really start things off by slaughtering a sacred cow.
A Cow Named Vance
I realize that many D&D fans don't mind Vancianism. For them, keeping track of the particular uses of spells and abilities for their characters and not being able to use them after a set number of times per day is like the air they breath. It's hard to even notice it as a thing, and if it's considered at all, it may even be gently approved. After all, it's a straightforward way to limit abilities that are too powerful for repeated use, and it's hard to imagine other approaches. What other options could there even be?
Before I answer that, let's define Vancianism. One of D&D creator Gary Gygax's favorite authors was Jack Vance, whose Dying Earth series became a model for magic in his new game. Gygax's approach was that wizards and other magic users would prepare a fixed number of spells each day, cast them once, and be unable to cast them again until the next day. It is as though each spell were a memory that casting the spell makes you forget. Usually in RPGs we think of "Vancian magic," but really any ability can be Vancian, whether mundane or arcane, if it has the words "per day," or "per encounter," or "before a short rest," or some phrase denoting uses limited arbitrarily by time.
It's pretty much the main way to limit power use in most d20 games. And it's simple enough idea, right? So why do I emit exasperated sighs every time I see the phrase "once before a long rest?"
This is where someone says, "We're talking about a game where wizards and Shaolin warriors steal treasure from dragons, and you want to talk about what's realistic?" But yes, we can absolutely talk about "realism" in such games, if we understand realism to mean something like "behaving in accordance with the tropes and genre conventions of fantasy fiction." Now obviously something like Vancian casting works in the worlds of Jack Vance. That's basically a tautology. But any other fictional mages work this way? Gandalf, Merlin, Baba Yaga? The hedge mages of the Black Company, the Aes Sedai of the Wheel of Time, whatever idiosyncratic magic users Brendan Sanderson* is working on this week? They may be limited by time, resources, or other means, but none of them find themselves simply unable to use a spell they cast five minutes ago simply because they cast it. (*Okay, the Mistborn could plausibly be modeled by something like spell slots, especially how they are implemented in 5e. But there are other problems; read on.)
It's even worse for non-magic guys. If a sword slinger is limited in the number of times he can use a particular maneuver, it should be because of tactical considerations, not because he hit an arbitrary cap. Yes, you could argue that such a cap is an abstraction of these considerations, but this is often unconvincing in actual gameplay, and there are often more interesting ways to model such things.
Maybe some people really like marking down every time they cast a spell or use some other per-diem resource. Maybe players of the 5e Paladin class, for example, find the epitome of playing a heroic shining knight of virtue is in counting spell slots, channel divinity uses, and hit points from their "Lay on Hands" pool. Maybe I'm grossly underestimating the appeal of such metagame accounting. But I doubt it. I suspect the reason players do this is because they know of nothing else, and would readily jump at the chance to try a better way if there was one.
Imagine entering the evil lich's lair, intent on upending his foul plans once and for all. This is your life's apotheosis, the moment you literally were created for (since you only really exist on a character sheet, after all). You raise your holy sword and utter a divine imprecation against this anathema as you prepare to smite him from this mortal plane. Only you can't, because you're out of smites for the day.
Vancianism says you can't do something cool because you've already done enough cool stuff today.
It messes with the narrative flow of a campaign
Perhaps you've heard of the "five minute adventure day." If a wizard runs out of spells, and the only way they can be restored is by taking a long rest, then he's going to want to take a long rest as soon as he can. The player is not being a jerk, or a min-maxer, or anything like that; he's behaving exactly the way a Vancian wizard would behave if he was thoughtfully trying hard to survive. This can really derail the sense of urgency that drives a lot of the sort of drama D&D is supposed to model. Of course, the DM can move the story in such a way that taking a long rest is impossible just when it would be optimal, but that's actually sort of cruel, plus I find that when the solution is "let the DM fix it," that's just another way of saying "we don't actually have a solution."
It's thoughtless design
I almost said "It's lazy design," but laziness implies knowledge preceding a preference not to act. If someone in the woods is starving but doesn't know how to hunt or forage, it isn't laziness that's keeping him hungry; it's ignorance. By thoughtless I actually mean "without thought," because with a few exceptions I really do suspect most d20 designers haven't even recognized these issues as problems, let alone thought about how to address them. And it's frustrating because there are lots of non-Vancian mechanics for limiting powerful abilities that designers could easily adopt if they ever thought to do so.
But, you ask, what might those mechanics look like? Patience, dear readers!