I'm not really sure I see the full problem here, either. 9th level spells break the game: yes. But how many games even get to 17th level? And at 20th level nearly every class is insane somehow. Druids pretty much cannot die from HP loss at that point; that's a fun one to compare.
For the sweet spot of >90% of games - levels ~3-6 - the two classes seem pretty even.
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Will casters start to dominate at high levels? Yup. Is that a playstyle that a lot of people want anyway? I think so.
I'm not sure going till 9th level spells is necessarily the best assumption. There has been a lot of work done about how long games go, and I think they said something like less than 1% of games even get to level 16. So I'd personally cap the levels at 16, as I think that's where the Tiamat storyline ends up (The first one is 1-8, so guessing second one is 9-16).
I am someone who like to run games to high levels (my 4e campaign is currently at 27th). So for me, it is a problem if a game's playability breaks down at those levels. And it's not as if it is especially hard to design the game so as to avoid this.
So why design the game in such a way that it does? I can only assume that WotC has market research data showing that more of the potential market prefers a game in which, at high levels, spell casters dominate over non-casters in capabilities and flexibility. From my point of view, that's a pity.
Remarkable athlete is not a fighter feature thought, it's a champion feature. And again, Champion is the choice of the options/complexity averse player. Now I know that making no options "balanced" with options was an explicit design goal, but it's an impossible one.
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The only way to avoid that is to rig the game so that at best the complex character can only hope to equal the simple character. Sound like fun?
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Could Remarkable Athlete be better? Yep.
On RA, I would be inclined to start with proficiency bonus to all physical stat checks. That would give more breadth than the rogue, but less depth (because the rogue gets expertise on some of them).
On simple vs complex, I think the "rigging" that you describe is closer to the way they should be designing, at east as far as core functions are concerned. The complex character should have more breadth, perhaps, or a greater ability to respond to surprises and edge cases. But when it comes to the core business of play (which in D&D is basically fighting things, finding things and talking to people) the simple class should be competitive.
As far as the 5e fighter is concerned, ways of handling this would be (just off the top of my head) some sort of "martial supremacy" option that gives advantage on rolls in combat comparable to the Foresight spell; a "remarkable leader" option that could be taken in lieu of remarkable athelete, and allow proficiency on all CHA checks; etc.
Referrring particularly to the bard vs fighter comparison, I think the bard being better at linguistics and performance is fine. It's the abilities the bard can leverage to muscle in on the fighter's core competence (fighting) that strike me as problematic.
I hope the above makes sense.
When someone says that "if one class can do X, everyone should be able to do X", it's not unreasonable to interpret that as doing the same thing from a mechanical perspective.
I actually think that's completely unreasonable. For instance, if someone says "If one class can break the game at level 17+, every class should be able to do so" then that doesn't imply the two classes should do the same thing from a mechanical perspective. For instance, perhaps one can summon a horde of demons, and the other can lay waste to an army or storm a castle single-handedly.
if a wizard can cast a spell that bypasses AC to deal direct damage and a fighter has a maneuver that bypass AC to deal direct damage, the only real difference is the name of the spell/maneuver and fluff. To me that's boring because there's no difference between the two other than fluff. And to go back to my niche protection comment, I don't like a game where every character type is just as effective as every other one in combat, at skill challenges, etc.
In this case, presumably you think the fighter should keep the AC-bypass manoeuvre and the wizard should lose it. Otherwise you have one character (wizard) which is better than another character (fighter) both in and out of combat.
The tendency of higher-level spells to eclipse fighters in combat is part of the issue that the OP is raising, I think.