I don’t think the parties DO run the primaries. At least they did’t here in VA, where the state ran them. The parties make the rules about getting on the ballot, but the polling places are managed by the cities/counties who report to the state.
Yeah, you are right, but it just doesn’t seem like there are enough checks and balances in place. ‘Rats can delay, manufacture and lose ballots and no one gets called on the carpet.