You can correct me, but I seem to remember that a lot of this open primary stuff had to do with the Democrat party in the Jim Crow South acting in the capacity of a private club, restricting membership, with the winner of the primary being the de facto winner of the election. So, any new party rules would have to make sure there isn’t even the appearance of any of that going on.
The SCOTUS (in Smith vs. Allwright????) in 1944 outlawed Texas Democrat Party rules that allowed a whites only Democrat Primary by party rule. That sort of racial discrimination having been outlawed in 1944, it remained so after 1972 when Mayor Daley the Elder successfully defended his delegation at the 1972 Demonrat convention by getting expedited review before SCOTUS to obtain a timely decision against then radical left Alderman Bill Singer and an alternative delegation of McGovern crazies. The rule now seems to favor party rules over state laws except in service to racial segregation.
Example, if in our home state of Connecticut, we had allowed Democrats or Unaffiliateds to vote in GOP primaries, Weicker would still be a Republican. He tried to change the GOP rules but we stopped him.