I do agree with Open Primaries, but we need to use approval voting for that. The regular runoff method still has the problems of vote splitting and you can get the two worst candidates in the general. With approval voting you get the two best candidates in the general election.
I think you're the first person other than myself I've seen on here recognizing this, and it is absolutely correct. Let parties have whatever "primary" process they want. Without control over the ballot who cares what method a PAC uses to pick its favored candidate? Like Strangelove said, prohibit anything on the ballot but the name of the candidate. If someone wants to vote based on their political organization's recommendations they can bring in a list of names to vote for.