Re: What do you think a Fighter should be?
See the problem is that Fighters are too generic AND too specialized at the same time.
They are very generic in that their main schtick is "Fighting with mundane weapons". This is extremely broad and can potentially swallow every non-magical class in the game. This means there are dozens of potential concepts you can apply to it, each of which are fairly different.
On the other hand, they are very specialized, in that "Fighting with mundane weapons" by itself is a very limited thing. You can hit stuff in melee, or you can hit stuff from range. Occasionally you can do something like push the enemy, or trip them. This is typically the sort of thing that gets accomplished by a single spell, and not even necessarily a high level one.
You can try to support the Fighter by embracing the broad concepts of Fighters, but rather than trying to make them specialize, let them be all of those things at once. Yes, my Fighter is a highly mobile swashbuckler, an amazing Archer, a stalward Knight, and a Leader of Men. Basically you have the Fighter go down the list of every mundane archtype, see what is acceptable for them to have, and give it as an option to the Fighter. Let him learn as many as he wants with downtime, much like a Wizard learns spells.
Going this route basically acknowledges how limited mundane characters really are when placed next to magic. Even when doing this, you're going to end up with a Fighter with a bunch of abilities roughly equivalent with 2 or 3 schools of magic, and that's assuming you are willing to embrace the Warlord style powers for him. The question is if at this point what you have is even a Fighter anymore. He is basically a Mundane Paragon. He embodies every martial class ever, because doing so is what it takes to compete with magic.
The other route is to, as suggested, break up the Fighter into subclasses, getting rid of the Fighter and instead have each of those concepts as their own unique class. Going this route, you'll have a bunch of radically underpowered very specialized classes, unless you are willing to let them have supernatural abilities within their niche. Even this is probably going to leave you with a bunch of tier 4 classes, because while you are letting the mundanes leave the mundane, you are still running with the idea of a bunch of specialized classes. Luckily, most of these specialized classes are a bit more broad than "Fighter" in terms of out of combat utility, so will fare a little bit better there.
The best route though, is honestly probably saying "Mundane classes last to level 6. Once you hit level 7, you must multi-class into something that isn't mundane" and being done with it.
If my text is blue, I'm being sarcastic. No, really. Don't bite my head off for saying the Fighter is way overpowered, or how Wizards need more spells per day to keep up.