> Suppose the Supreme Court should decide that the Supreme Court should decide who can and cannot run for president
They do no such thing, they decide questions of law. Article II eligibility is a question of law. The Electoral College performs a ministerial function, they do not decide questions of law.
Of course, we can remove any doubt by just beginning with a conclusion. If we begin with the assumption that measuring the qualifications of a presidential candidate must be a judicial function that can only be performed by the judiciary, then we will conclude that it is a judicial function that can only be performed by the judiciary.
If, on the other hand, we begin with the assumption that measuring the qualifications of a presidential candidate must be an elector function that can only be performed by electors, then we will conclude that it is an elector function that can only be performed by electors. We should not begin with a conclusion.
There is a legal question as to WHO under our Constitution is empowered to determine the qualifications of a presidential candidate and the judiciary can decide that legal question, but we should not assume that the Supreme Court must decide that question by ruling that only a court can determine the qualifications of a presidential candidate. In fact, there is good reason to believe that the Supreme Court, if it decides that legal question at all, will decide that the electors decide with finality the qualifications of a presidential candidate and that their decision is not reviewable by a court. If the Supreme Court really wanted to decide that legal issue as you believe they should decide it, then they certainly had a recent opportunity to do so. As you will recall, they made no effort to get involved.
So, maybe we shall soon see. You may be right about what the Supreme Court will do, but I believe that you are wrong. I don't think the justices want to be looking over the shoulders of electors and I do not believe that they want to get involved in DNA tests or comparing photos of putative fathers and sons or hearing testimony of who was sleeping with whom or engaging in any of the baloney that we saw in the social media during the last eight years. It is inherently political.
There is nothing wrong with the procedures that we have followed for more than 50 presidential elections.