I
run 5e.
Concentration is a high cost that makes a caster often choose between offense and defense leaving them very vulnerable to attack.
It's a meaningful enough restriction on the spells that require it, sure.
...the caster can't stack spells to kill himself.
Lol. I'm sure you didn't mean it that way, but it's still funny.
And the concentration mechanic. And Legendary Resistance. And Bounded Accuracy making it quite possible for a creature to save.
Have nothing to do with versatility.
5E has toned this down considerably from 3E. Far more than you give it credit for.
I give 5e full credit for being less broken than 3e as far as class balance goes. 3e was, however, the most broken ed of D&D, by far. Even the experimental early rulesets weren't as radically class-imbalanced as 3e. 5e is probably vying with 2e for second most-broken class balance in an edition of D&D. Even so, if 5e were simply following on the heals of 3.5, that could be seen as an improvement. But it follows Essentials, which followed 4e, so the class-balance trend is sadly to the negative. 5e's saving grace is that it's very up-front about leaving the tools out for the DM to fix those issues in play.
What characters are you talking about? And what do they do that can't be accomplished by the fighter?
Really, even the most basic things: surviving in the wilderness, providing leadership, negotiating, dealing with kings. Fighters have just been profoundly limited outside of combat in D&D. 5e Backgrounds help a bit in that they make everyone more well-rounded, but that doesn't excuse the fighter class for being so narrow.
There are other classes to mirror different types of martials. Fighters are fine in 5E. They are highly effective. They are not severely outclassed by casters.
Not /severely/? Maybe not. But all 5 of the arguably-martial sub-classes in 5e are functionally similar, in that they contribute little or nothing beyond DPR in combat. The fighter is unique in how little he contributes outside of combat. If you combined the fighter and rogue into a single class, with the DPR of the fighter, and skills and out of combat goodies of the rogue, you still wouldn't have a broken class.
They even have a few abilities that are nice in a non-combat sense like the Champions proficiency in all physical skills.[/quote] half-proficiency, anyone proficient is better than the Champion.
The Battle-master's ability to assess opponents.
5E fighter is one of the most balanced and capable I've seen in D&D.
It's one of the worst-balanced and least capable out of combat I've seen in D&D. Which isn't saying anything different than what you're saying, really, since the fighter hasn't varied wildly over the last 40 years, and the worst-balanced (the 0E, probably, maybe the 3e, though that's unfair as it's the casters who were the source of imbalance) and the best balanced (4e fighter, again, mainly because of the other classes), respectively are all we're excluding. The 5e fighter is fairly typical of the narrow strengths and broad failings of the D&D fighter: high damage, low contribution outside of that.
Not sure what more you want unless you're pushing for anime style capabilities. I would prefer that not be in the game myself.
Calling broader or more fantastic abilities 'anime' doesn't make them undesirable. The alternatives are really to bring the fighter up to the excess-of-genre level of D&D casters, or pull casters down to the shy-of-genre standard of the fighter, or strike a balance closer to genre. If you don't want "anime-style" fighters, then you should settle for sub-genre-powered casters, the kind that have only a trick or two, if any, that's combat-useful, and mostly provide exposition and the occasional plot-important ritual or item-based magical mcguffin, that would balance with that.