I think there's a subtle difference that you are disregarding. The Twenty-Second Amendment says…
No person shall be elected to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once.
It doesn't say that such a person is ineligible to be President again, only that he is ineligible from being
elected President. I just don't see anything in this wording that prohibits him from being elected to any other position, which would put him in the line of succession to the Presidency, nor of assuming the Presidency if those ahead of him in that line are unable to do so.
And the Twelfth Amendment does not say, as you claim it does, that one who is ineligible to
run for President, or to be
elected President cannot run for Vice President. It says that
“…no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.”
You are assuming that being ineligible to be
elected President means that one is ineligible to hold the office of President. Your assumption seems logical on its face, but given that there are other ways to become President than by being directly elected into that position, the wording of the Constitution just does not support this assumption.