Quit thinking like a lawyer. Grand LG
It's actually covered under the 12th Amendment as you stated... sort of:
But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.
Which speaks to who can actually hold the office. The thing is, an argument could be made about a legal loophole... The 22nd speaks to the
election of a President and Vice President. Constitutionally, Obama cannot run again.
"Elected to the office of the President and Vice President..."
It does not address whether or not a former President could be nominated and confirmed to the office of Vice President (in a Vice Presidential absence) since any vacancy is an act of the Senate. Which is a technicality since there is no election involved for that position.
Hypothetically, Joe could have had Kamala removed on day 2 of his Presidency, nominated Obama to fill in as his VP and it would go to the Senate for a confirmation, they turn around and either 25th Joe or he resigns and, Barack Obama becomes the 47th President of the United States.
Without ever being elected just like Gerald Ford. So, technically Obama
could assume the office of President since he meets the Constitutional requirements to hold the position (35 years old, US citizen, 14 years a resident) but not having been elected. Hence, no violation of the 22nd Amendment since he wasn't elected to the office.
Now, I'm pretty positive the SCOTUS would intervene and have to rule on such a thing, but then they start talking about the "spirit" of the 12th and 22nd Amendments which opens up it's own can of worms.