Unlike the basically ceremonial office it is today, until the late 19th century the US Presidency was both the Chief Executive and Head of Government. Until US Grant's term. Sam Tilden OTH and his successors seemed content to let the legislature make the policy, which led to the situation as elucidated by the great Constitutional Scholar and President, Woodrow Wilson* that but the gist of all policy is decided by legislative, not by executive, while the legislature is guided by the United States Council; the Speaker, the minority and majority leader in both house and the Vice President.
Could the US have maintained a Presidential form of Government, or was the office of President, like the Kings of England, destined to lose power?
How would it change the relations between the Federal Government and the states? Congress has expanded its powers jealously. Can't see the Commerce Clause being used so expansively.
Maybe the VP would be elected on the same ticket, instead of being elected by the Senate due to the whole "electors don't cast votes for VP" convention.,
OTH, the Presidency would be for serious politicians, not retirees like LBJ** , worthies like Eisenhower or prominent citizens like the current holder, Donald Trump***.
*OOC He did actually say that since in the years after the Civil War the prestige of the office fell until TR revised it.
** Man was a great Majority Leader and VP, but frankly after his heart attack following the 1957 Civil Rights Act, the Blair House was the only place he was going
***Well he was the price the Senators Clinton demanded when it seemed clear that Pence would get the VPship.