Congress exercises a ministerial function in counting the electoral ballots, and such ballots can be discounted only if the certificates are not in regular form or are not authentic, the role of Congress is limited to problems with Electors and does not extend to the eligibility of a candidate (3 U.S.C. § 15). Only in a special circumstance may they exercise any other power (Amend. 12). In no circumstance do they possess authority to determine who is and who is not eligible for the Presidency.
The role of the Electoral College is limited to casting votes in accordance with the popular vote of the State that each Elector represents. Their function is a ministerial function. In no circumstance do they possess authority to determine who is and who is not eligible for the Presidency.
Neither Congress nor the Electoral College decide questions of law. The Judiciary has exclusive authority to decide questions of law.
You did not find those notions in our Constitution. Under our Constitution, each state decides how it shall choose electors and those electors vote for president and vice-president. The Founding Fathers did not believe that states should be required to choose electors by popular vote. What you have written is what you apparently believe the Founding Fathers should have written. Lots of people believe that we should get rid of the electors and let the president be chosen by popular vote. But, we haven't done that yet. The electors are still choosing the president and vice-president. I believe that they are obligated to vote only for candidates who are qualified to serve. So far as I know, every elector has agreed with that view of their responsibility.
We're just going to have to wait to see if the Supreme Court wants to become a screening committee for presidential candidates. I don't think it will, but again, we'll have to wait to see. ;-)