The parties usurped the voters' role in the choosing of the vice president in 1832.
We're a weird country. Few electoral democracies elect a president the way we do and even fewer deal with succession the same way. Most nations replace their departed presidents with a temporary fix, typically an acting president who is a parliamentary official analogous to the Speaker of the House pending a special presidential election, or a quickie election to find a replacement. We're pretty much on our own when it comes to figuring out a better construction.
What's clear is that nothing would be lost and much would be gained by requiring presidential candidates to declare their running mates, and their cabinets, up front.