So, we have primaries for each party. Some states are open (anybody can vote, but only for one party's candidates) while other states are closed (only party members can vote). Then, after the primary winners are selected, there is a general election where to select among each party's winners.

Most of these elections are plurality, although some states require majority (with runoff between 2 top candidates).

Basically, this leads to extreme candidates winning the primaries, then trying to pivot to center for the general election. It also means that if a primary has many options, as the GOP 2016 primary did, you get results like Trump - all the reasonable candidates cancelled each other out.

Pivoting to approval voting would be relatively easy - the ballots don't change much (just allow 1+ choice per race). We could maintain party/state primaries and a national general, but move them all to approval. That would avoid 3rd party spoilers and hopefully prevent the fringe of either party becoming the nominee/winner.

This ignore the added complexity of the Electoral College. That's an anachronism for another post.