It's my understanding that to be the president you have to meet three requirements.
Mrs. Albright fulfills the second two but not the citizen requirement.
Was this not planned for when she was appointed?
What would've happened if Clinton was in a NATO country (destroyed) during a nuclear strike and Gore and Congress were annihilated in a decapitation strike while she was on a humanitarian mission somewhere?
Would she become president without meeting the citizenship requirement?
Or does the line of succession not care about country of origin?
Both the 1886 and 1947 versions of the Presidential Succession Act specifically pass over someone who is either not Constitutionally eligible or has been impeached. There has never been a requirement (and I'm pretty sure none has ever been even so much as tentatively proposed in Congress) for Cabinet officers to have been born in the United States, which would have ruled out some rather talented people from Alexander Hamilton and Albert Gallatin onwards.
You may be interested in my prior answer on the 1886 version; I need to go add Part III of that response at some point.