I think a large problem with primaries is that, in most states, you have to be registered with the party before you can take part in their primary. Sure, you can game this by switching registration before and after, but it's still very limiting. The POTUS would be more representative if the primaries were not allowed to do that. Sure, it means that one side can try to vote for a joke candidate to make it easier that their own candidate wins the office, but it also means that "honest" voters can pick a candidate from the other side that they would be most likely to support (or least likely to oppose, anyway.) If that person does get into the office, it's still involvement from more of the population in doing so, which might help fight the "us vs them" mentality in US politics.
Of course, the Democrats and Republicans both bank heavily on that very "us vs. them" mentality to rally their base, so I wouldn't expect change in the primaries, electorate college, or first-past-the-post voting short of a revolution (political or physical.)