Well, that's not what the amendment says, "if the President elect shall have failed to qualify....." The amendment lays out the procedure to be following but it does not give congress the authority to make that decision.
Who do you suppose has the power to disqualify the President elect?
I have no idea because neither the constitution, nor congress has made that clear which is not to surprising since most Americans in our early history rarely if every traveled outside the country. The issue never came up until 1880 when there were rumors that Chester Arthur, the vice president nominee for president was rumored to have been born in Canada. In 1964, Barry Goldwater, the republican nominee for president faced qualification accusations because he was born in the Arizona Territory. In 1968, democrats questioned the qualifications of George Romney, a republican presidential candidate born in Chihuahua, Mexico. Senator John McCain, republican nominee for president faced questions because he was born in the Panama Canal Zone
It seems to me, the simple solution would be for Congress to make the Federal Election Commission responsible for certifying that a candidate met the constitutional eligibility requirement. If the candidate failed to qualify, he or she could take the case to federal court. In this way, the issue could be settle before the first primary and the campaigns could focus on real issues that concerned voters.
Five other presidential “birther” controversies from American history