Dear 100 Hour Board,
I've heard that people who are ineligible to become President can't be on as vice President because of ineligibility of succession, but farther down on the succession list is cabinet member Elaine Chao who is ineligible by non-birth-citizen status, and as such, would just be skipped over. When does the "you can have this position, we'll just skip over you" start? Do you have to be eligible to be president before you can be speaker of the house or pro tempore of the senate?
The only 2 positions where they won't let you have the position for non-birth-citizens are president and vice president. Speaker of the House and Pro Tempore of the Senate are both elected by the House of Representatives and Congress respectively, with no restrictions as to who is eligible. All cabinet positions are available to any citizen regardless of non-birth-citizen status. If you want to read more about the line of succession, the wikipedia article is quite in-depth and talks a lot about the nuances and history of ye olde line of succession (source)
Hope this helps!