I'm afraid you are still not grasping what the author is saying. Nowhere can i find him saying that serving under a President makes on inelible to succeed him. That is preposterous. What the author IS saying is that one can't serve as VP if he/she is not eligible to serve as President. That is 100% correct. Again, this piece is not about cabinet members. It is about Presidential and VP eligilibity, period.
2nd paragraph:
Why? Because she cannot constitutionally fill the vacancy caused by the death, resignation, impeachment and removal of any US president under whom she served.