Stacking extra attack

[Skrum]

You know there was a D&D edition that did what you describe. 4e, and my understanding is 5e has solidly trounced it in terms of popularity. It's possible when designing games in general to be too balanced, and the D&D player base has straight up proven it's okay with or prefers a level of imbalance if it means they get more variety with their characters.

While I have not played 4e extensively, my general thought is that 4e general unpopularity has caused WotC to run away from a lot of good ideas that 4e was on to.

Martial classes actually having a proper progression and powers that exceed "guy at the gym"? Yes please.

I simply refuse to believe that accomplishing the following is impossible, or that these goals are somehow at odds
1) a martial ability and power progression that ~keeps pace with spells up to level 20
2) unique and diverse character classes

IMO, I think gearing classes more towards short rest/encounter based recovery would be beneficial to accomplish the above two goals (and some others besides). But it is not a goal unto itself.

My back of the napkin ideas for each class?

Wizard = looks something like the psion from 3e. Runs on spell points, lots of spells known. I would favor spell points coming back on a short rest, and spell points would cover spells up to level 5ish. 6th and up spells would be more like warlock's mystic arcanum; know a handful of spells that can be cast 1/LR

Sorcerer = looks something like the wilder from 3e. Runs on spell points, few spells known, but can metamagic/overcast the spells for increased effect. Their t3 and t4 progression would come from more and more powerful ways to modify their spells

Cleric = basically a divine wizard, but would gain weapon and armor proficiency (as is custom) and would give up some spell points. They would keep channel divinity, which is a great feature. High level spell casting would be similar to wizard; they'd use "miracles" to perform great feats

Druid = wildshaper class. Would have limited spellcasting that looks kinda like current warlock (without mystic arcanum)

Bard = warlock-like spellcasting. Expand uses for bardic inspiration, bake more weapon and armor proficiency into the base class

Rogue = bring back the factotum!!!!

Fighter, Ranger, Paladin, Barb, Monk = they'd all have their own class features (paladin is a particularly good example of a martial done right), and each would progress along a "martial path" a la Tome of Battle. Fighter would have access to the most/all schools, while the others would be more focused on schools that suit their class flavor. Paladin and Ranger would no longer have spells

Obviously I'm just spitballing. But none of these classes feel samey at all (well, except maybe for cleric and wizard), and there's plenty of room for them all to be powerful.