In the recent IL and NH ballot challenges, the elections commissions were split on that point. IL said it had the power to rule on the merits, NH said it was not competent to rule on the merits. The IL ruling was "summary," did not discuss the objectors argument at all. Just picked one side, and said that's it. Totally flawed procedure by the election commission.
I come down on the side that election commissions and secretaries of state are comptent to rule on the question, in order to protect the integrity of the primary ballot. If the parties don't like that, they can quit conducting primaries. First presidential primary in the US was 1921 or so, IIRC.
If you read the decision of the Illinois Ballot Commission you will see that it quite extraordinarily brief, a couple of sentences, after the finding that standing existed on three points. What is even more unusual, so far as its value as any kind of precedent, it does not cite a single piece of authority and consists of a mere unsupported conclusion. As precedent in would worthless in an actual court. The Illinois law indicates that an appeal lies directly from its decisions to the Clerk of the County Court and from there there lie several layers of appeal, including ultimately to the SCOTUS. Whether Joyce will take an appeal is not clear.