At the time of the election, when the House counts the votes it is implicitly approving the eligibility.
This argument can't hold under the Constitution. IF the person elected president were, in fact, ineligible *under the Constitutional standard* (which no one seems to know what that is, exactly), nothing the House -- or anyone else -- did or could do would make him eligible. This is not a procedural defect that can somehow be overcome. Like if it were later determined that the president was only 22 years old when he was elected, there's no way he could be deemed to have met the age requirement.
If the SC got involved on this part they would be extending their power over an issue that is not given them by the constitution.
I don't see how this holds water either. The eligibility standard is set out in the Constitution. The SCOTUS always has authority to determine if an act of government violates the Constitution. All the usual rules, of course, would apply -- there would have to be a case or controversy, jurisdictional requirements must be met, and the party bringing the case must have standing to do so. The latter seems to be one of the major procedural snags. Who has standing? I'm not familiar with how whatever complaints have been filed were disposed of by the Court.
My read is the constitution distinguishes between a president elect and a president.
It gives instructions regarding a president elects eligibility and it gives separate instructions for a president that is not able to serve. For a president, it no longer gives eligibility requirements. In fact, it is possible that the succession rules provides us with a president who would not be eligible to be a president elect.
My read is that once the president is elected, there is no longer an eligibility issue because he is already the president.
The House appears to be given control over a president elect and a president. First, by counting and approving the electorate votes and then by impeachment. It also provides a solution if the president elect is ineligible but is silent when a president is ineligible. It also specifically limits impeachment to the House. If the SC began to determine the president did not have power then it would be effectively impeaching the president, a power specifically limited to the House.