The ancient Roman republic (i.e. pre-Caesar) did just fine run by the senate without any emperor (they were vehemently opposed to the idea of putting so much power in the hands of one individual).

One of the ways they made this work was by assigning temporary powers to elected/appointed individuals appropriate and sufficient to get their job done, whether that was provincial governor, tribunus plebis, consul or whatever. A notable title was that of "dictator" (where the modern word derives from) which gave sweeping temporary powers to a general sufficient to execute a war.

Julius Caesar famously abused this system, effectively becoming the first emperor (despite no such position existing) by assuming the title of "dictator perpetuo" (perpetual dictator), although that did not end well for him. His successor, Augustus (whose adopted name later became the name of the office of emperor) initiated a more acceptable version of this by simultaneously holding multiple "elected" positions thereby having the power of them all.

The Roman Republic was incredibly decentralized with more people working for the administration of the city of Rome than for the Roman state itself at the time of Caesar's conquest of Gaul. Extreme executive power existed, but not at the level of the imperial administration. Provincial governors had near absolute power and generals had control over their armies to a degree well beyond what any general today has. They were basically private mercenaries with the generals, not the state, being in charge of payroll for the soldiers. Local territories had near 100% control of their own affairs so long as they did not do anything to interfere with the tribute flowing to Rome.

This became more centralized over time and as it did more executive power was needed at the top level. Augustus centralized the army under state control. Emperors in the late empire had nearly dictatorial authority and a massive state bureaucracy under their control.

> 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)