But no electoral college would mean it's much more viable for third party candidates to run and there's no reason why you'd have to stick to the current system with first a primary inside each party, then the two party's candidates running against each other with virtually no chance for any outsider. It does of course make a difference what system precisely you'd use - ranked choice voting? A multi-round voting system like the French presidential election in which tactical voting can play a strong role? And how does each party organize its primaries beforehand? Etc.

Another big factor would be whether the congressional voting system remained the same in the meantime or not - as long as Congress remained locked in the two-party duopoly, that would make it harder for presidents who don't have close links to either party. Though not impossible - there have been a number of cases of independent/third party governors in various states who've also had to work with state legislatures stuck in the two party system, it would be a similar situation to that.