AND JUST WHY ARE NON-REPUBLICANS ALLOWED TO VOTE IN ANY REPUBLICAN PRIMARY?

I hear what you are saying.

An open primary allows Democrats to vote for a weak Republican candidate, and hurt the Republican chances for victory.

This cuts both ways, though. Republican voters can do the same to the Democrats.

Years ago, I lived in Pennsylvania, with closed primaries (I don't know if that's changed). The problem was that I was an Independent, and was not permitted to vote for either a Democratic OR a Republican candidate.

As an independent, I was disenfranchised from the primary elections. I am not the type of voter who just votes for the Party candidate, no matter what. Historically, this is a poor strategy. A strong conservative Democrat may be better than an unprincipled liberal Republican, for example.

So, open primaries serve a purpose. It allows independent and unaffiliated voters to have a say in the primary elections.

California employed a semi-open primary. Concerned with "tricks" by the opposition in this largely Democratic state, the Republicans closed their primary to Republicans. Independents could vote for Democrats or other candidates. Unfortunately, there were no other candidates, so the independents were forced to vote for a Democrat.

I think fear of tricksters are overblown. If I were to cast my primary vote for the weakest candidate from a party (as a game, strategy, or trick), and that candidate won the general election, then I participated in a fraud on myself and on the Nation. I get what I asked for. That ought to make potential tricksters think twice.

I like your objection. What if Party's were allowed to be closed, but Indies could vote from any of the candidates? That way parties could not have outsiders tamper with their results, but they could consider how many indies voted for one of their candidates in their selection process.

This gives Indies a chance to influence decisions, but also prevents one party crossing over to mess up the other party's candidate selection process.

I am just do stream of consciousness here--haven't thought this out. But there surely is a way to keep one party from screwing with the other party's selection process as I believe Dems did with Republican process in New Hampshire, but also allowing true independents to be able to participate.