> Aren't the primary elections the business of the parties conducting them and not really part of the democratic process at all

That's the ideal but not the real-world truth, where the parties are entangled with the legal process. If a person leaves one of the major parties, it is almost impossible for them to be recognized legally as a candidate for a presidential election. That's one of the reasons why you get Communists and Centrists in the same political party. The can't possibly ascribe to the same party platform, yet they are members of the same party.