The Constitution is our second frame of government. We had Presidents under the first frame, too.
See 4
We did not have Presidents under the Articles of Confederation.
Remember it was a confederation of states that each had a governor and had been running their own affairs for years during the war, no new super-executive position was thought necessary at the time.
Later events led people to believe that an executive would be necessary and that is one of the big reasons that the new constitution was written and adopted.