I can offer a start on the problems with Roman government. For specific reasons Americans chose an elected president we'll have to wait for someone else.

The Senate is just a name; it's not at all like the Roman Senate. The Roman Senate was comprised of ex-magistrates - they had to be elected once, but then were in the Senate for life. Ours doesn't have term limits, but requires elections every six years.

About the rejection of two consuls, I know the founding fathers (I hate using that term - they were not a monolithic group) were reading all kinds of political philosophy from Locke and Montesquieu to Polybius, and the concept of the mixed constitution was present in all of these (this originated in Aristotle, but others developed the idea), that is the best constitution is part monarchy, part aristocracy, and part democracy.

Romans divided the monarchical portion between two equal powered consuls, and in the later stages of the Republic this hamstrung government often and often brought things to either disaster in the field when two consuls refused to work together (Battle of Arausio 105 BCE) or government shutdown (such as Caesar and Bibulus in 59 BCE). The problems of the two-consul system would have been completely apparent to the framers (still feels like an icky term).