Contesting the electoral vote and the 20th article are two different things entirely. If a states electoral vote is challenged successfully then that’s states electoral votes are ignored. So you would have to object on some kind of grounds enough states to swing the count below the required amount.
The 20th simply states “if the President elect fails to qualify by the beginning of his term...” In other words there is supposed to be a process by which one would offer qualifications but because we have two “clubs”, NDP and NRP, who make their own rules we are hopeless in trying to force this issue.
The 20th Amendment provides Constitutional grounds for someone with standing to challenge whether a President elect (one who has been certified as the electoral winner) has “qualified” by Jan 20th. I believe that a member of Congress would have standing.
At any time a party who has suffered particularized harm from any of Obama’s appointments, orders, laws signed, or nominated SCOTUS justices would have standing to challenge his ability to “act as President” given that he could not have qualified by Jan 20, 2009. That’s where somebody like Hobby Lobby comes in.