That's my point - this person is claiming that serving under a president makes one ineligble to succeed them. And that's false.
And the gender issue has been resoundingly refuted by others. Suffice to say that when the Constitution mentions members of the House, it refers to them as "he" - but we currently have female House members, with no legal challenges to their service because of their gender - which means the writer's line of reasoning is completely irrelevant.
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.
I wouldn't be surprised if some crank with nothing better to do has in fact gone to court to make this goofball argument against a female Congresscritter who had annoyed him somehow. If the judge also had nothing better to do that day, there may be an actual legal precedent formally declaring that the argument is without merit.