There seems to be a bit of truth in it. An unaccountable head of state is much more dangerous than a Prime Minister you can be removed by the parliament. So let's assume the Founding Fathers recognize this danger, but instead of introducing a Prime Minister or introducing the Impeachment, they decide to form a dual leadership on the Roman model.
Is this possible? And how could this be achieved?
Given the fate of the Roman Republic, I'm not sure they would be so enamored of the idea.
The irony is that the Federalists at the convention basically shaped the US constitution to be a republican version of the British system as practiced in the late 18th Century. In other words, the president was supposed to be more powerful than the British monarch of today but less powerful than the president today.
Over the years, the British system saw power flow from the monarch and the Lords to the Commons and the cabinet. The US saw exactly the opposite, with the Senate in some ways growing more popular than the HoR and the presidency growing fairly dominant.
