Actually "Why do we need an eldritch knight when there's a wizard?" We don't - Arcane caster, Divine caster, Fighting man and Rogue are the main classes. Everything else is a mixture of these.
You just renamed it.... but i can use your words too.
Why do we need an arcane caster if we have an eldrich knight?
Having 4 (5,6) superiority dice per short rest is perfectly fine. Every special ability in 5e is limited. If you wanna disarm, trip or heal every round... I reckon you picked the wrong edition.
You can already disarm and trip every round. Those are basic actions anyone can take. Heck, a fighter can trip 4 times per turn.
No one is asking to heal every round. Though there are several ways to heal every short rest.
Though Everyone can get damage reduction each round, and there are a few ways to get THP each round.
But, let's do some math.
If you consider that a eldrich knight is 1/3 wizard, then you can consider the battlemaster to be 1/3 warlord.
Thus a full warlord would have 3x the dice, giving you 12(15, 18) dice per short rest.
Divided by 2 combats * 3 round each = 6 rounds of combat per short rest.
= 2 (2.5, 3) dice per round.
2 dice per round is balanced. Going to 3 dice at some point, like level 11.
And that still leaves room for a few extra features. Like letting someone spend a hit die.