That's a really weird definition of "political party". I don't know of any country that has political parties by that definition.
I'm not aware of any law that requires primary elections. As I understand it, that is a decision by the two major parties.
But I don't think it would help this cause either way. The author wants a party to be able to enforce ideological purity. That's going to pare your party smaller and smaller, until it's a party of one.
If ideological purity is your goal, then you can just write yourself in to every election. And then you will spend a few years under the governance of somebody who was able to make compromises and get a plurality of votes.
I don't think the law is this guy's problem. But even if it were, he'd immediately run into a million other problems.