Nick,
That would be true if we Elected the president by popular vote. But we do not. We Vote for Electors who Elect the President.
Also the Constitution clearly states the US House of Representatives is responsible for Certifying or determining eligibility.
Even though people vote for electors, the people still have the right to know on what basis the electors have determined a candidate eligible. The fact that we elect an elector is not to admit that we are too stupid to be worthy of the vote and to know on what basis a candidate is considered eligible.