The features probably need to focus on effects other than damage.
So, what about those options I wrote: lift over 4000 lbs, run over 40 mph, leap over 100 feet?
Super-strength seems archetypally appropriate, from Norse Thor to Marvel Thor. A temporary hulking out for 21+ giant strength works, mainly for the carrying capacity but Strength bonus to damage too.
I've thought about this a lot today while at work. One issue is a lot of such things seem more appropriate to barbarian or some other class than fighters...
For example:
Powerful Rage (or Mighty Rage). When you rage, your Strength score increases by an amount equal to your proficiency bonus. The increase lasts as long as you are raging.
So, with Primal Champion, while raging, a Barbarian could have STR 30 (base 20 + 4 for primal champion + 6 while raging with powerful rage).
A similar feature would allow you to act one (or more) size category larger when raging. Combining the two ideas would allow a Barbarian with STR 30 and the lift capability of a gargantuan creature.
But I have a harder time seeing that for a "Fighter"....
Additionally, "subtle" spells, such as slot-9 Foresight (gain advantage / impose disadvantage) and Invulnerability (damage immunity!) might be appropriate.
Mechanically, sure, but how to justify the narrative without it seeming "magical", which is something A LOT of people want to avoid when bumping up the Fighter.
For me, the challenge is coming up with a supporting narrative for such features. I've had little success for those cases with Fighter.
Now, again, for other martials not so much. I could see a revised Foe Slayer:
Foe Slayer (level 20 Ranger feature). Your knowledge and understanding of your favored foes is unsurpassed. You gain advantage on attacks made against your favored foes, and your favored foes have disadvantage when attacking you. You also have advantage on saving throws made against your favored foes. Any Intelligence or Wisdom check you make concerning your favored foes cannot roll below a 10.
Perfect Defense (Monk feature). You can spend 1 ki point and enter a state of perfect tranquility. For 1 minute, you have immunity to all damage, but must spend an additional ki point at the start of each of your turns to remain in your trance. (up to 10 ki total)
Dont forget to make the appropriate sacrifices to swap in these powerful Wizard class features. Even so, they allow the Fighter to function at the level of slot-9 spells.
Yeah, there has to be some trade-off. Whether it is limited use, duration, or some "sacrifice" as you say...
An acceptable way to magically empower a Fighter is to grant the Fighter a magic weapon or armor. It is easy to make a magic item part of the design space of the Fighter class. When the Fighter character levels, let the player pick which magic item one wants. The narrative can be awkward, how does the nonmagical Fighter acquire this magic item? The magic item might be bound to the fate of the Fighter, and appear when the Fighter is ready. The Fighter might be a kind of famous sword forger, who can imbue the weapon with a part of oneself, or magic the weapon by becoming one with it.
Maybe, but again the narrative is always important to me as well.