"RG can judge" is rather arbitrary and the minute someone got screwed by such an arrangement, you would be the first one demanding changes.
Yes use judgement. Sometimes Cabinet officials have to use judgement. The Secreatary of Elections having to determine if the intent of voter is clear on a weird ballot. Some people also want the SoFE to decide what is "campaining in the voting booth". It can be very subjective. The Registrar would not accept a request like the far fetched example of one party voting for expulsion of members of another party or some random request of one individual targeting another individual.
Expulsions are rare anyway. Nobody got screwed in this case. The RG can judge a request made by the Chairman is valid. It followed the internal rules of the party, members had time to vote, the result is clear, the result id not contested. The RG and the public can see a party decided it wanted to cut links with one individual.
I am not a member of a party and NCYankee is chairman of a major party so I can't believe I am defending the right of political parties. The Federalist is a recognized party. It can set its rules. There is a provision for expulsion. The party has the right to expel someone and not be associated with this person anymore. That would include not being listed as in the party, not running in elections carrying the party's name. Like a political restraining order.
The other view that is presented is a citizen can linked itself to a party and the party has no right and must endure the situation. A citizen can run in election carrying the name of the party even if the party does not agree. You could have two presidential tickets from the same party (one not supported by the party) creating confusion with voters. A citizen using the party's name could behave badly in public or do illegal things soiling the reputation of the party. And even after an expulsion the citizen could rejoin the party in an endless loop. Is this the absurd reality?
I think parties have the right to decide who is in the party, who can use its name, who can be associated with it. They have the right to expulse. they have the right to tell the RG it doesn't want an individual to be associated with it and stop listing someone as part of the party following a reasonable process of expulsion.
I don't think any of that legitimizes the majority altering the registration of a single member. This is basic equal protections and no, political parties are not clubs. They are composed of registered members, the rules for such registration being under the authority of Congress to regulate, as per the constitution. If we maintained our own roster of members and had complete control over it under "our rules", then yes, it would be entirely up to "our rules". Though ironically, "our rules" aren't much help in this situation either for a variety of a reasons already stated. But our list of members is maintained by the Registrar General, in accordance with the laws. People are removed from that list (deregistration, falling off the roles, in accordance with provisions established by the constitution and statute), so that list is regulated and protected by Federal law.