No, literally any qualifying candidate can run in any primary, and parties don't decide who's qualified -- statutes do. I'm not sure why you seem to think that parties get to decide who can run in their primaries. They can't. They can decide who they will back, but they can't keep people from running in them because they don't like them.
Now you seem to have gone from 'allowed Bloomberg' to run to 'allowed him to speak'. I don't know if you're moving goalposts, but this is pointless.