What do the nine references to “eligible” in Section 19 refer to?
They refer to the people who would “act as President” if the elected POTUS can’t act as President. The title of that section is “3 USC § 19 - Vacancy in offices of both President and Vice President; officers eligible to act”. That part is not about the electoral certification. It doesn’t mention Congress’ certification of the electoral vote at all, and has no connection to anything mentioned in the previous sections.
It is saying that if there is nobody who is able to “act as President” (for instance, if the courts had found that BOTH the President elect and Vice-President elect had “failed to qualify” and were not able to “act as President”), this is the order in which a person would be offered the ability to “act as President”, IF THEY QUALIFIED - meaning they are “not under disability to discharge the duties” of the Presidency. Being ineligible to be President (not NBC, not at least 35, and not a resident for at least 14 years) is a Constitutional DISABILITY.