Would there even be a President as we know it?
When the Convention was writing the part about the Executive, it did so with a clear awareness of who would hold it. Without Washington, do they go for a plural executive, perhaps with one man elected by the Senate, one by the House and one by the States, or keep a one-man executive but with much more limited powers, with the Senate choosing judges, Cabinet officers etc? Perhaps the President of the Senate gets the ceremonial functions like receiving foreign ambassadors etc but has no veto on legislation.