> The ancient Roman republic (i.e. pre-Caesar) did just fine run by the senate without any emperor

If you don't count the 80 years of off and on civil wars and political and social upheavals.

> and sufficient to get their job done

Well they were extremely successful military, internal politics were a mixed bag and it slowly turned into chaos and constant gridlock as soon as they ran out of external enemies to fight.

Rome had an arcane deeply adversarial political system which basically relied different powerful men in the elite competing and keeping each other in check while trying to amass as much wealth and political power they could get away with (Ceasar was hardly an aberration besides just being way more successful than countless aristocrats playing the same game).

> effectively becoming the first emperor (despite no such position existing) by assuming the title of "dictator perpetuo" (perpetual dictator),

Sula did that first, in many ways Caesar just emulated him (of course the decision to not murder all of this potential enemies came back to bite him)

Sulla even managed to resign and survive, which I think is even more impressive.