Actually, fifteen years ago I was making the case that Bill Clinton could be Al Gore's VP, and I made the same case in 2004 regarding a Kerry-Bill Clinton ticket.
XXII does not make a person "ineligible to the office" in the XII Amendment sense, it makes them "ineligible to be elected" (whatever that means).
Gerald Ford wasn't elected.
I agree with you that the language is inartful but I continue to think that the problem is greater than language, it is a threat because our body politic is very sick and vulnerable.
In other words, language in the long run is not the problem and, certainly, language will not spare us if we are headlong on a course of self-destruction.