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The XXII Amendment could have and should have repealed and replaced the XII Amendment, adding prior service to the "qualifications to the office" section - but it didn't.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?
Hillary Clinton/Barack Obama Ticket To Replace Joe Biden?
That cannot happen because of the 12th. Obama is not eligible to run as either president or vice president.
Can someone appoint him as VP (subject to approval by congress) if the elected VP should resign or die? Yes.
But even so, he cannot stand for election for VP when the president who appointed him VP wants to run for reelection.
A lot of these amendments have not really been challenged in court as far as I know.
As I recall, Theodore run for president again in 1912, after serving 3 years of McKinley's presidency then winning in 1904 to serve another 4 years, making 7 years altogether, then taking 4 years off to do other stuff like hunting in Africa. But then that was long before the 22nd which became law in 1951 if I recall correctly.
But who's name is going to be on the VP slot for the elections? If they put Obama’s name on the ticket, The Supremes will vote it down.
Only way I can see it is if they put a stooge as VP, who then resigns after being elected and Obama is then nominated by The President.