And even in states with closed primaries, lots of people register to influence the other mainparties candidates because their party can’t/won’t win at the state level…

This. Those that do this got to vote against trump twice: Once in the primary, once in the general. Also “registering” in some states is simply saying the name of the party when you vote in the primary. You are given that party’s ballot, and your name shows up as “registered” for that party.