In small words it says "whether one wishes to know about science or religion, there is a place for teachers."
Religious indoctrination is not teaching. There is a very important difference between science and religion: One is evidence-based, and the other isn't. In one a teacher shows you things you can then do for yourself, and build upon. In the other, you are asked to believe with no evidence or proof. In one, if you can show your teacher was wrong and a different answer is better, you are a hero. In the other... well, fortunately the times where you'd be burnt at the stake are past.
Religion and science could not be further apart. The fact that people talk about them, including teaching and "teaching", is about the only thing they have in common. Don't confuse them because of a random correlation.
I beleive he was taking a more inclusive view about the abstract hierarchy of church leadership and the preisthood in general rather than the very narrow view of the Vatican specifically.
He said "Vatican", not church. He certainly meant the institution and not the plot of land, I'll grant you that. From the way he consistently spoke about the Vatican, not the church, I would not assume he means something he doesn't say. Even within the catholic church, the Vatican is a special case.
Really, If the GP only meant the Vatican specifically then his point would have been specious.
Would it? I'm pretty sure the old men who enjoy the pleasures of having their own tiny state would disagree violently. There have been a few power-struggles within the catholic church regarding the position of the Vatican over the past decade or two. I don't think that would happen if it doesn't matter.