These are primaries for the purpose of choosing a party’s nominee, not a run-off designed to narrow down the field of candidates.
Even in states with true open primaries (where you can vote in any primary regardless of your affiliation) you can’t vote in more than one party’s primary.
I’m not saying our party system is the best possible system and I honestly don’t know enough about voting patterns to say whether open or closed primaries are better. But if you’re going to have parties with primaries as opposed to a big all-candidate free-for-all with a series of run-offs, then there’s definitely logic to restricting participation in the primary to those who have committed to registering as a member of that party.
You don’t technically register as an independent. But if you register to vote and don’t check one of the party affiliations on the registration form, you’ll effectively be registered as an independent (technically “no party”).
