Let’s be clear here. Major-party primaries in the US are absurd.
Parties in most countries select their nominees privately and pay for it on their own dime. That’s not surprising either considering that political parties are private organizations.
In the US, however, parties do their primaries publicly. That may sound nice, but as private organizations, US parties get to make their own rules. These rules conveniently include who gets to vote and how those votes are counted.