My diagnosis of the fighter is a little different; I believe the main problem with the fighter class is it's try to satisfy at least two very different character fantasies - and each of those could support one or more classes by themselves.
The first is the more classic Broadly Competent Fighting-Man (or -Woman). This is the fantasy novel hero like Conan, Aragorn, Sam Vimes, Mat Cauthon (unless you consider him a rogue with a fighter dip) etc. They're good with weapons - all weapons. They can pick up a sword, bow, spear, axe, mace or any other weapon known in the setting and go mess up some mooks. They may have a favorite, but the favorite is an individual weapon that means something to them personally (and may be magical).
They probably can't cast spells, but they can use magic in other ways like magical items (not just weapons), simple rituals, or maybe they just know a bit about magic so they can think of counters to it.
They're also broadly competent at other warrior stuff - they can survive in the wilds, track, tame horses, search for traps and disarm them, lead troops, organize a barricade, etc. Soldier stuff or similar, but they actually have quite a few skills. They may even have a few skills that sit outside the normal warrior/adventurer list, but not too many.
The champion looks like it could do this, but you never get the skills for it.
The second type is the specialized weapon master. They have a single type of weapon that they want to truly master - they might be proficient in other weapons, but they'd rather use a regular sword than a magic axe if they're a sword-master. They could pick any weapon but swords and fists are the most popular in fiction. For a vanilla example of this trope, look at Kirigaya Kazuto from Sword Art Online, though I would add in Lan Mandragoran offhand. It's more popular in anime than western fiction, but lots of players have watched a lot more anime than they've read pulp fantasy.
These characters may or may not use magic, but if they do they use a lot. 4e swordmage levels of magic use, if they go this route. Others disdain magic for pure sword skill, but the skill they achieve matches what magic could give them.
They generally don't have many skills beyond weapon skills - maybe something related like calligraphy or military science, but no more than one or two.
The battlemaster was supposed to handle this, but falls extremely flat when used that way.
I think in a DnD-like game, you cannot put these concepts into the same class and make it work (without majorly changing the role of classes generally). The first needs solid weapon proficienies, lots of skills, and good defenses in or out of armor. The latter needs a list of specialized techniques to learn, significant bonuses to specific chosen weapons, and a resource pool to fuel their bursts of awesomeness.
The first could also be a nonmagical ranger. You could split the latter into magic and non-magic classes (ie sword-saint and swordmage) to allow for different resource pools, but the two will play out in similar ways.
Leadership options (a la marshall/warlord/cavalier/PDK) could also be a thing - but I'm not convinced it makes for a good class in a game with 5e's chassis. I think I'd rather add a "warfare" skill and some specific uses like "motivate an ally," maybe with feats, and let people add that to whatever character they already have.
