I think what would be interesting is a semi-open
primary.
By this I mean that all the parties's candidates are listed and every voter may vote for one, and only one, person.
This means that there couldn't be the vote in one primary, change parties, vote in another — but it also offers more freedom than your voter registration says 'independent', you can't vote in any primary
.
There's nothing wrong with that.
If you want to vote in a primary...pick a side.