I don’t find Professor Posner’s reasoning sound. To be vying for election but to not be able to be President if elected seems like a tangible harm that is not speculative. To go to all the expense and effort of becoming the President only then to not be able to be President is not in the least less than a real prospect of substantial harm. Judge Posner is highly regarded but his son’s reasoning in this brief article seems quite flawed.
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.