Its not circular at all.
The 22nd says a two term president cannot be elected to office, therefore they are “constitutional ineligible” for Presidential office.
The 12th amendment says anyone not constitutionally eligible for the presidency is not eligible for VP. You cannot have someone in the VP role (whose entire purpose is to assume office in the event it’s needed) that could not legal assume the role under their own eligibility.
The entire framing of the 22nd was to limit presidential terms and prevent another FDR situation from serving more than 2 terms (10 years). You can’t circumvent the intent of the framing through word play (ie ”elected”) to allow a former president from repeatedly assuming the presidential office through a VP role.
I understand your thinking, but until this actually happens it's hard to say how the courts might treat this (if at all). Most feel that Congress are actually the arbiters of who is or isn't qualified to be President.
However, here's a Congressional Research Service paper that touches on this very subject:
Does the Twenty-Second Amendment Provide an Absolute Term Limitation on Presidential Service?
The Twenty-Second Amendment prohibits anyone from being elected President more than twice. The question has been asked, however, whether a President who was elected to two terms as chief executive could subsequently be elected Vice President and then succeed to the presidency as a result of the incumbent’s death, resignation, or removal from office. Another version of this scenario questions whether a former President who had been elected twice could succeed to the office of chief executive from other positions in the line of presidential succession, such as the offices of Speaker of the House of Representatives, President pro tempore of the Senate, or positions in the Cabinet, as provided for in the Presidential Succession Act.
This issue was raised initially during discussions of the Twenty-Second Amendment in 1960, when President Eisenhower was about to become the first President covered by its limitations. While the question may have been largely academic with respect to Eisenhower, due to his age and condition of his health, it was also raised again concerning former President Barack Obama,
Some commentators argue that the Twelfth Amendment’s statement that “no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President” ipso facto bars any former chief executive covered by the Twenty-Second Amendment from serving either as Vice President or succeeding to the presidency from any other line of succession.
Others maintain, however, that the original intent of the Twelfth Amendment’s language was only to apply the same qualifications of age, residence, and “natural born” citizenship to the Vice President as apply to the President, and that it has no bearing on eligibility to serve as President. Moreover, they maintain that the Twenty-Second Amendment’s prohibition can be interpreted as extending only to eligibility for election, not service; by this reasoning, a term-limited President could be elected Vice President, and then succeed to the presidency to serve out the balance of the term. Adherents of both positions, however, generally agree that anyone becoming President under any of these scenarios would be prohibited from running for election to another term.
Assessing a related question, legal scholars Bruce Peabody and Scott Gant asserted in a 1999 article that a former President could also succeed to the presidency, or be “acting President” from the wide range of positions covered in the Presidential Succession Act. By their reasoning, a former President serving as Speaker of the House, President pro tempore of the Senate, or as a Cabinet officer would also be able to assume the office of President or act as President under the “service vs. election” interpretation of the Twenty-Second Amendment.....
It seems unlikely that this question will be answered conclusively barring an actual occurrence of the as-yet hypothetical situation cited above. As former Secretary of State Dean Acheson commented when the issue was first raised in 1960, “it may be more unlikely than unconstitutional.”
