You may want to look at the 12th.
See my post 86, I cover the XII Amendment issue.
The fact that a man cannot be "elected" to the office does not make him "ineligible", there are a number of ways to become President without being elected, Lyndon Johnson was not elected on November 22, 1963 (for example).
And, since Presidential elections are not mentioned in the Constitution and are unnecessary for the State Legislatures to perform their core function of appointing the Electoral College, it's not even clear who or what are burdened by the XXII Amendment.
Is it voters? Is it State election commissioners? Is it State Legislatures? Is it the Electoral College? Is it the House of Representatives, in a case where they must choose the President? Is it the Special Joint Session of Congress where electoral votes are counted?
The XXII Amendment could have and should have repealed and replaced the XII Amendment, adding prior service to the "qualifications to the office" section - but it didn't.
And as a practical matter - if 270 Electors for Vice President of the United States send ballots naming Barack Obama to the Secretary of State of the United States on December 15, 2020, and if those ballots are opened and counted by the Special Joint Session on January 4, 2021, do you doubt for a second that he will be sworn in as Vice President on January 20, 2021?