First off, I want to say that the following comes from the perspective of someone who has grown estranged from 4E and doesn't like Pathfinder. I started with 3E and grew dissatisfied with its many flaws, but neither of its replacements really suited me.
I sympathize with you greatly.
To state it crudely, I hope 5E doesn't have a Fighter class. Or rather, I hope it doesn't repeat the mistakes of some older editions and try to create a single class that is somehow supposed to embody all the concepts of "a skilled warrior who fights without the aid of magic". All that does is create a generic, flavorless chimera of a class that inhibits the development of more interesting class concepts.
However, I very much hope you or someone like you isn't in charge of 5E design.
To speak freely, I think you've become cynical and disspirited. Now, I can't blame you in that exactly, but I do want to say that just because you've seen obvious problems go largely unsolved and ignored for years doesn't mean that no solution exists or that we must settle for something which, while perhaps fine in itself, is perhaps a solution to a problem other than the one we had in the first place.
I like seeing all kinds of different warrior variants. Heavily armored knights, archers, swordmasters who forego the use of armor, lightly armored cavaliers, etc. I hope 5E's rules leave room for all of these kinds of concepts and let different character types shine.
There is absolutely no reason why these things can't be done in a single interesting well done fighter class. The tools are there, but among other things it requires a less hesitant, passive and conservative skills system than 3e provided. Tumble is an example of where they could go with skills having potentially relevant roles in combat. It also means accepting that skills beyond a certain level represent superheroic and not merely mundane ability - Batman's abilities at dodging out of rooms unseen, taking long cuts to end up ahead of those he chases, jumping from incredible heights without taking damage, and so forth.
I also hope that 5E continues some of the good ideas from the Tome of Battle and 4E by letting such non-magical characters still perform impressive feats and use advanced combat techniques.
You mean by letting non-magical characters still perform magic. This is the cynics solution to the problem, and I reject it.
and creating class-independent fighting powers similar to 3E's arcane and divine spell lists could be really interesting.
Notice that we keep trending towards, "Well, if only spellcasters can be awesome sauce, we ought to let everyone be spellcasters."? But note, that the real problem is, "Everyone should be able to be awesome sauce."
Older editions seem to have always marginalized such rules based on the idea that mounts don't belong in a dungeon...
I agree with you that unusual play styles shouldn't be marginalized, but in older edition's defense, part of the problem is that a mount usually
is more trouble than they are worth in a many dungeons. The real problem here is the assumption that dungeon delving is the core activity of the game, and so other varieties of play don't need full support. The game should let you play the game you want to play, whether its Cossacks on the steppes of the 'Real World', or fantasy dungeon delvers in a quasi-medieval tolkien-esk consensus fantasy settings. If your campaign takes place on an virtually endless open plain, mounts are going to play a big role and the rules should support that. If you are in a maze of twisty passages, all alike, then it should support that to and not insist that, because the game has mounts that they must be just as useful (Poke-mounts) in all situations. This suggest one of the reasons I dislike the notion of a 'mounted class' and other narrow and inflexible concepts.
