In Canada (where I'm from) our party leaders are elected by the party members. Whenever there is a change in leader, there is some party convention and paying members of the party are asked to vote. Anyone can pay to be a member, but it's not a public vote like in the states. I wonder if this produces better or worse candidates? With our parlimentary system, we also don't vote for our Prime Minister, we vote for our local representative.
I think it's obvious if the US worked this way, the Republican party wouldn't have selected Trump. Perhaps you end up just electing the same old boring white guy time and time again.
In the US, voting is decentralized (state control their own election/ballot processes). The parties control their own candidates.
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.
I think it's obvious if the US worked this way, the Republican party wouldn't have selected Trump. Perhaps you end up just electing the same old boring white guy time and time again.