Great post! I've long thought about the same thing. One thing I started leaning on in my thoughts is that the Fighter is the regular person. No, I don't mean they should only get mundane stuff, I mean that Fighters represent the every day NPC. They're the one people can see themselves in the most. The other classes have social baggage (rogues are thieves, paladins are high and mighty, barbarians are wild and uncivilized, rangers are one of them there rangers...).
This draws up on old Fighter tradition of becoming knights and lords.
The other think I've been thinking about is tweaking the fighter subclasses to mean something other than mechanics. Just with the PHB subclasses:
The Champion should be the people's champion. Gladiators, war heroes, beloved nobles, infamous villains. The Champion should be loved or feared, but above all be respected. They should have some charisma abilities.
The Battle Master is the artist. The Way of the Warrior is an art (hence, the tool proficiency, like samurai who practiced calligraphy). They should be insightful, collected, wise, and should have wisdom abilities and abilities of perception and insight of people and other warriors.
The Eldritch Knight... Well they're intelligence focused but I'm not sure where they should be in the world. They could have come from a tradition of guardians of wizard academys, or stem from the elven tradition of blade magic. Leaning into their bonded weapon, they can be the warrior that does magical things with their blade, and could get unique spells like cutting so hard a wave of force goes out.
The Psychic Warrior is in a similar situation. They could be jedi. They could see their warrior training as a religion, or they can just be the warrior that hones their body and mind to supernatural levels. I propose changing their name to "Immortal", but that's just because I find Psychic Warrior to be uninspired.