You are incorrect. Obama's own defense used the Twentieth Amendment, Section Three as their reason to disallow standing to those who demanded to see proof of eligibility in court. Their argument was that Congress was charged with this responsibility under the Constitution.
It makes no sense to me that Obama’s attorneys would try to use section 3 of the 20th Amendment. Do you happen to remember which lawsuit that was?
If Congress had a role in presidential eligibility determination, I wonder why in the six plus years since February 10, 2007 when Obama first announced his candidacy, not one second of congressional hearing time has been devoted to his eligibility?
Now I do see Congress having a potential role in blocking the certification of an ineligible candidate’s electoral votes, but that’s about it. A president-elect can “fail to qualify” under the 20th Amendment by not having 270 or more electoral votes certified at the Joint Session of Congress that is held to count and certify the votes of the electors. Any one Senator along with any one Representative can submit a written objection to certifying any state’s electoral votes but there were no written objections in 2008 or in 2012.