There in lies part of the problem as I see it. It may demand qualifications but no state or the Feds have set up guidelines/laws/protocol for establishing the demanded guidelines such as what must be provided, to whom, when, where, etc..
Did you read the argument in the link? This is also in the U.S. Code. The current procedure which forces a congressperson to “object” is actually unconstitutional if you study what the twentieth amendment says. I would suggest that those “in line” to succeed the President if he “fails to qualify” MUST be presented with evidence that he is indeed “qualified”.
" The Senators and Representatives before mentioned, and the Members of the several State Legislatures, and all executive and judicial Officers, both of the United States and of the several States, shall be bound by Oath or Affirmation, to support this Constitution;"