And no, I don't want wuxia.
You sure, your examples sure sound like wuxia to me. I've pointed them out, below, in case I have misunderstood.
I'm more concerned with the extremely static nature of the fighter and rogue. They're the same as they level up, just with bigger numbers. Going from +5 to +11 with your athletics checks over the course of 20 levels is just embarrassing.
Well 5e uses "bounded accuracy" so I don't think its fair to knock the system for giving you +6 bonus on Athletic checks.
She should be strong enough to rip the arm off of a troll and shatter admantite shackles.
Wuxia IMO. A wizard casts a spell, sure it is more powerful but it has limitations - finding the spell, learning the spell, components, range, duration, concentration...etc
What you are speaking about is an innate ability, doesn't sound all that limiting. Once again this option may be included in the DMG where you swop out an attack for a Wuxia power. Thing is
BASIC should not have wuxia-style powers (and it doesn't). It's
BASIC.
Legendary magical items like Exacaliber and Gae Bulg should be drawn to her, and only someone like her should be able to use them to their full potential.
This is a great idea, but should rather be a requirement of the magical item rather than the fighter class.
She shouldn't have any trouble turning a village into a powerful miltia, some of which will volunteer as her personal followers.
This can be done during roleplay - you can argue the same about a Rogue and his Thieves Guild, or a Cleric and his Church moving into position of power (Bishop, Archbishop...etc), or a Wizard and the Mages Guild of the Land - apprentices seeking him out...ANY CLASS.
Unlike the wizard who needs to spend most of their treasure on studying magic just to keep up, the fighter should have plenty of gold for parties, castles, and a wide array of magical items.
I've lost you here. Why can't a wizard find treasure, gold, abandoned castles, magical items, spells through adventure? Is the fighter the only one able to? What am I misunderstanding? How would you bake this sentence as part of the class features?
A fighter should be able to outrun just about anything, and when they jump, the main limit should be their sight, not their legs.
Sounds like Hulk and therefore Wuxia, refer to my above post regarding limitations.
All that said...you can have a fighter that does all of this and you still wouldn't break the game
The reason they are not included is not because they "break" the game, its because they limit a large play style, right out the box.