I feel like most of these changes would require a constitutional amendment.
Why was this downvoted? Yes, it's likely you would need at least one amendment. The Supreme Court has ruled that the government cannot impose open primaries on the political parties. See California Democratic Party vs. Jones: https://www.oyez.org/cases/1999/99-401 This decision has been affirmed more than once, and it makes sense.
In such a system private groups would still be free to endorse a particular candidate.
Won't happen, but the problem isn't that primaries are closed, it's that they impact the names that end up on the ballot.