me the biggest counter to this statement is: Are fighters "at high levels" happy?
Entirely depends on the DM and their chosen adventure design (among other things).
All the classes, especially at high level, suffer when they aren't being given enough to do concurrently with each other, and Id argue given my experience playing at high level (my group actually almost exclusively played at this level for extended periods because we did not diddle daddle with the XP) that its an even bigger issue than intra-class disparities.
People often point to Wizards being able to shut down entire encounters with one spell, and while I aggressively agree that this is terrible design,
at the same time, if an encounter is so shallow that one spell by one player is enough to end it, then it was just a bad encounter to begin with.
The thing about game design is that just shutting down the obviously broken isn't the only thing you need to do to resolve such issues. A shallow encounter is still shallow, even if theres no "end this encounter" button anymore.
When you approach the issue from this perspective of resolving both, that also helps to temper what you change so that they're not only less drastic (and thus less likely to induce unforeseen consequences), but require much less work to keep balanced relative to each other.
So sure, as Ive said elsewhere in a different way, martials can and ought to have more to do in and out of combat, especially at high level, but the encounters themselves need to be better too.
Ive talked about this example before, but one of my favorite encounter designs is the siege of a mountaintop city by an army of hundreds (1000s) of Orcs, a cabal of 9 Arch Mages, and an Ancient Red Dragon. While it functions around homebrew horde mechanics, what it accomplishes is sufficiently occupying a group of 6 level 20 players, without any players being able to just shut down the encounter, or going without the opportunities to be special.
Between protecting the town, slaughtering the army, dueling the mages, and fending off and eventually slaying the Dragon, theres plenty to do and a lot to consider about how to do it. Having better designed classes in this context would be cherries; delicious delicious cherries, but its never quite as satisfying to just eat Maraschinos out of the jar when you could have a whole Sundae.