The main principle of all political institutions of the Roman Republic, applied during several centuries, was that in any executive function 2 or more people must be elected and a single one can be accepted only temporarily, during emergencies like wars or other calamities.

There are a lot of modern countries who claim to be republics and even USA has a "Republican" party, but all these modern uses of the word "republic" are wrong, because none of them uses this principle of filling all executive positions with multiple people, so that none of them will have all the power of those positions.

The modern states which use the word "republic" justify their claim by having elected leaders, but that was not something specific to the Roman Republic. There were plenty of ancient states with elected leaders and with various other lesser public functions filled by elections.

What distinguished the Roman Republic from all others was the requirement of electing multiple people for all positions, including for the supreme executive function, for which they had 2 consuls, and not 1 president.

> I'm saying there's no logical reason why we need to vote for one person, rather than say 3.

There's many logical reasons. The federal bureaucracy handles the day to day work of federal agencies, not the actual individual elected to be the President. The President appoints a cabinet to oversee those agencies. So there's no need for any individual to be awake 24/7 making decisions which means there's no need for multiple shifts of executives to be available.

Multiple executives would actually be a terrible system as situations last longer than a single shift. Day Shift President would have to hand over a bunch of context to Swing Shift President for any issue lasting longer than a single shift. Swing Shift President may not agree with Graveyard Shift President on the best course of action which leads to an inefficient and/or legally arduous process of adjudicating primacy of authority.