I have a better idea
Return selection of Senators to the houses of each state, and the selection of the President and Vice President back to that senate.
Problem solved.
I Concur.
The President and Vice President were never selected by the Senate. It has always been through the Electoral College, at least under the Constitution. The Senate does select the Vice President if the Electoral College doesn't have a clear winner.
Perhaps you meant what the Electoral College was suppose to be. With it essentially nominating the candidates and the House of Representatives (voting by states) electing the President from those finalists.