Adopting a parliamentary system would avoid many of these problems—albeit it would require a constitutional amendment. The President could be elected by the House of Representatives from among its own members, for an indefinite term, but could be sacked by the House at any time by a simple majority vote. The President would continue to be a member of the House, and automatically lose office as President if they lost their seat in the House.
America’s federal system makes snap elections difficult. And I don’t think anyone right now wants to centralize our elections.
I'd go further and say that the rest of the cabinet should also be elected from the House (not just the President). This ensures a cabinet member is answerable to both the House and to an electorate. Though the two-year terms in the US House are probably too frequent for a parliamentary style system like this (perpetual campaigning).