Er...I'm not sure that that would change anything.
New registrations of "Democrat" or "Republican" wouldn't county any more, and people registering as such would have to check out all the parties.
Sure they would.
Party registration is not required by law, it's just a precedent everybody keeps following.
Only username and home state are required.
Which gives me an idea:
Repeal "organized party" status - we never really filled it with meaning anyways.
Have the SOFA stop publishing people's party affiliation.
Party membership to be created (for future registerers) by joining at party forums or the like.