I think if you want more realism, if you'll pardon the abuse of the word realism, in the game it makes more sense to switch to a more skill-based set of rules anyway - GURPS, Hero, White Wolf, JAGS, etc... Because levels are a similar kind of arbitrary framework that don't make sense inside the game but make things easier at a meta-game level.
I do in fact like skill-based systems, and back when D&D3 was new, I loved that it looked so skill-based (but it wasn't, really). But skills also have their limitations. Sometimes it's fun to play with the more class-based specialized roles. Skill-based systems sometimes lead to characters that are very similar, whereas class-based systems force specialization.
But realism isn't really the same consideration as diegetic/non-diegetic. In general traditional skill-based systems tend to be diegetic, but actually almost all traditional systems are diegetic, including most editions of D&D. The abilities on your sheet, the decisions you make as a player, are all directly related to the abilities of the character and the decisions the character makes. 3/3.5 started the move towards more abstract abilities, and 4e and Pathfinder definitely continued in that direction. Pathfinder definitely has some abilities that are too abstract, to unrelated to something real that the character might do in the world he lives in. And that makes it less diegetic. 4e goes in my opinion too far overboard in that direction; a cleric needs to hit an enemy to heal an ally. The fighter can do a fantastic attack for tons of damage and a cool side effect, but only once a day, whether it's the only fight that day or they're going to have 10 more fights that day. What's keeping him from doing it again? It makes too little sense from an in-character perspective, and the abilities and limitations feel very artificial to me.
But skill-based systems can also have non-diegetic elements. In FATE, for example, you can choose to have an aspect used against you, because it gains you a fate point, which you can later use for something good. It's a simple way to reward players for having characters with weaknesses and roleplaying them, but the decision to succumb to the flaw because it gets you a fate point is not the character's decision. But this mechanism is more separated from the character's actual abilities. It's the same meta currency for everybody regardless of what's on your sheet, and you'll never suddenly stop being able to do something.
The particular problem with using critical hits in Dungeons and Dragons instead of some sort of meta-currency is that the critical hit system that has existed in every edition before 5 isn't that potent. I haven't read about the critical hit rules for version 5, maybe it's improved. Critical hits in earlier editions are not useless by any stretch, but even if your DnD3 fighter uses a lance and has Weapon Specialization, and is against an opponent that is not immune to critical hits, he's got at best a 15% chance to do triple damage. (Maybe 10% or 20%, I don't remember.) It's better than nothing, but in a climactic battle scene it can't hold a candle to the effects of a cleverly targeted Charm Monster spell or Fireball or Improved Invisibility.
10% is pretty good, actually. That gives you a pretty decent chance of a critical hit every big fight. Of course if crits aren't powerful enough for your taste, that's simply a matter of tweaking the system. It's not an inherent problem with crits by themselves.
Not even counting Dungeons and Dragons 3/3.5, even in earlier editions with spellcasters that were weaker than their 3e counterparts, at high levels they still handily outdid Fighters for combat utility.
That is unfortunately an inherent problem to the quadratic way in which spellcasters rise in power. They get more spells, and every spell they have becomes more powerful. That last aspect has been reduced a bit in 5e; now Magic Missile only produces multiple missiles if you use it in a higher level spell slot, which would cost you the use of a higher level spell. That way the new spell slot they get is more powerful, but the old ones don't also increase in power, which I hope does a lot to mitigate their quadratic growth.
Meanwhile, fighters get a slightly quadratic aspect. They get their specialty dice which get better as they level up, and they get more abilities to use them on. To what extend this balances the classes, I have no idea. Time will tell, I guess.
So if you wanted to replace meta-currency with critical hits or something else equivalent, critical hits need to be improved. Or possibly a Fighter's base attacks and damage need to be improved, so he doesn't need better critical hits to be awesome.
That's something that Pathfinder did. Pathfinder fighters can do crazy amounts of damage. In a recent campaign I played in, the 7th level fighter did 2d4+20 points of damage on a regular hit, and double that on a critical. And with the right feat, his chance of scoring criticals will double, so that's a pretty massive amount of damage. But ultimately, Pathfinder still suffers from the quadratic wizard and mostly linear fighter problem.
I grant that in theory in Dungeons and Dragons and in practice in many other roleplaying games, all characters do not need to shine in combat. As long as every character has a fairly common chance to shine at something, the game should be fun. However, Dungeons and Dragons has had a very strong combat focus since the earliest editions.
That always used to be my impression too, which is why I avoided D&D for a long time. However, due to recent exposure to OSR, I learned that old school D&D combat should be more strategic, rather than the more tactical combat of 3e and later. And by strategic I mean that you don't stand there bashing the opposition until one side collapses from accumulated damage; instead, you try to figure out a trick to sway the entire circumstances of the combat in your favour. You avoid fair combats at all costs. Changing the combat could of course be due to a spell (it's what battlefield control wizards do, after all), but it could also come from equipment or anything else. I once heard of a group that drive a herd of sheep into a dungeon to set off all the traps, for example. But it relies more on player creativity than the abilities on your sheet. It's the old player skill versus character skill debate.
But even in tactical combat, strategic goals can make the combat more fun and give lousy combatants something to do. I remember a Pathfinder combat where we attacked the guards of a prison wagon, while my Rogue tried to pick the lock to free the prisoner before their reinforcements arrived. That's fun stuff. I don't need to kill anyone in that kind of combat, because I'm doing the thing that actually matters.
Basically, combats that are about grinding through the opposition's HP are boring. You need ulterior goals. Do you need to free someone? You you need to get away? Get past them? And is combat really the only way to accomplish that, or could you try talking or sneaking your way past? In reality, nobody stands and fights until they drop. People flee when the combat doesn't go their way. I believe older editions had morale rules, which unfortunately disappeared when 3e came along.
So even if we remove the magic-sounding encounter and daily powers from 4e from the newer editions, it's likely that 50+% of gaming sessions is a battle. The person playing the cleric that only heals well and especially the one playing the thief that only steals well are going to spend a lot of time twiddling their thumbs waiting for lucky dice rolls in order to feel useful, while the fighter and barbarian (and especially the wizard) are going to be highly effective.
But that's a matter of adventure design. If the adventure is only about combat, then non-combat abilities don't matter much. Suddenly everybody needs to be a fighter in disguise. Of course it's also possible that it's simply that the combats take along time to resolve. This is definitely true in 3.5 (and even more so in Earthdawn), and according to my limited experience with 4e, combats there aren't exactly lightning fast either. Older editions were much faster. Thing is, the more detail you put in combat abilities, the more time it takes to use them, the more combat slows down, and also the more you want to slow combat down in order to get the chance to use those abilities.
And of course, it's worth mentioning that Earthdawn turns this problem on its ear by making every PC class explicitly magical. So the Fighter can have magic-sounding cool powers because he's actually using magic. I loved the game concept, though Earthdawn seemed a little too complicated to me. I never actually played, though, maybe it works better in practice than it appeared on paper. I keep hoping Earthdawn will get ported to another gaming ruleset, but no such luck.
Earthdawn sometimes gets called "D&D done right", because it has a real in-world explanation for all the weird unrealistic stuff in D&D. Various Disciplines (classes) were actually surprisingly balanced, despite spellcasters having no limit on how many spells they cast or know. But it's absolutely complex despite its elegance. Many Talents were practically their own subsystem, and combats could be unbelievably slow.