The secretary of state is in charge of the state elections. The primaries are state elections. In Cal she had the responsibility to see that the candidates were qualified. It is the Nov election for which she accepts the candidates that the parties supposedly qualified. A fox and the chicken house process.
See 652, the right track would be to look at individual state statutes to find out how to removed sec of state for violation of oath of office.
You seem to have missed this statement in the post you are replying to.
"In fact, as another freeper has shown, the CA Secretary of State is compelled to put the major party candidates on the ballot, with no discretion allowed. So that's not the law either."