If it operated like the Roman Republic, the Senate would not technically be involved in the day to day running of the state at all, except in the religious sphere. The state would be run exclusively by elected magistrates, who could be advised by the Senate and who would enter it once their magistracy was completed. Just to nitpick there.
Very true. The Senate's power did come from it having political authority in the modern sense. The Senate technically had no authority. A magistrate could ignore the Senate to his heart's content until his term ended. See C. Julius Caesar and his invasion of Gaul completely against the Senate's wishes or any of the numerous individuals (such as Marius or the Gracchi) who tried to go around the Senate and were only stopped by violent Senators, not any power of the Senate itself.
As I see it, and obviously my view is quite simplified, the Senate is like one of Picard's staff meetings on the Enterprise. The opinion of the senior staff mattered, but at the end of the day, Picard made the final decision. The opinion of the Senate mattered, but the magistrate at the end of the day made the final decision. The key difference is that Picard knew he would be captain for the foreseeable future and Riker couldn't do much about it. A magistrate was only in power for year (or a few years for governors and certain generals) and one day VERY soon someone he annoyed in the Senate would either be in power and against him or would sue him for his illegal actions.
The Senate's power came from its institutional memory and the force of its personalities. That and the fact that its members usually wanted to play by the rules and would do what is needed to keep people in their place.
A dictator and a senate couldn't really run Ancient Rome, let alone a modern country, on a long term basis. An America with Roman-style magistrates though would be an interesting creature. The problem is that with such limited terms, the best I could see is that ATL America evolves something like they have in San Marino: the elected leaders are symbols while their cabinet "advises" them as a cabinet would "advise" the monarch in a constitutional monarchy.
Could such a country stay together in the 18th and 19th Centuries? And if it stayed together, would this alt-America be able to become a great power with such a potentially chaotic system?