Sure nominate anyone, but who makes the final decision on eligibility?
If everyone is responsible," then no one is responsible.
The Democratic Party takes the position that it can nominate any crook it wants to - and yet it is entitled to have its candidate have a chance for victory. Thats precisely what happened in New Jersey, when the Dems renominated Sen Robert Torricelli even tho he was known to be a crook. After the cutoff date set by NJ law for replacing a nominee, the Dems realized that the torch couldnt win, and convinced the NJ Supreme Court that they were entitled to have a candidate on the ballot who could win.
The 20th specifically states, speaking of vacancies in the P of VP office, If a President shall not have been chosen before the time fixed for the beginning of his term, or if the President elect shall have failed to qualify