Alexander Hamilton favored a monarchy. George Washington was kind of ambivalent. When a presidency was established he wanted the title of the office to be "His Mightiness". Glad that they settled on "Mr. President."

If we had gone to a monarchy I think it would have been something similar to the British experience of a constitutional monarchy. At that time the power of the monarchy had been seriously circumscribed by Parliament. The Stewart dynasty ran afoul of it and Charles was executed for treason when he tried to bypass Parliament.