Actually, the Electors should choose the President.
The whole voting for committed slates of electors required by their state's laws to vote for whoever got a plurality in a Presidential popularity contest in their state (with the happy exceptions of Nebraska and Maine, where that applies to two electors, and the others are bound by the popularity contest outcome in a Congressional district) is a subversion of the Founders' intent.
We'd be better off if the electors were selected by the upper house of each state's legislature, or perhaps two selected by the upper house of the Legislature, and one by a popularity contest in each Congressional district. Or better still it we had people run for Elector without the absurd spectacle American Presidential campaigns have become, then once the Electors get together -- yes, make the Electoral College actually meet -- let them discuss possible Presidents and Vice-Presidents and maybe have a procedure for trial votes before the one that actually is recorded and either elects a President (and Vice-President) or sends it to the House.