The speech they're from doesn't.
The speech defends and praises universities and their role in society. Vance even claims some academics prefer to ignore evidence that refutes their positions, and he's against that; that would be a valid pro-intellectual position if true (but it's completely nebulous and unsourced)
The thesis of his speech was he doesn't like the content of what academics profess and he thinks they ought to teach his political views (and his audience's political views) instead. That's not anti-intellectualism, i.e. "don't trust those book-learning types, look to the common man for answers". This guy still wants ivory towers provided his cronies are in them.
Also it's interesting to see where his quote came from. He clearly picked an on-theme Nixon quote just to appeal to his audience, and he seems to miss the context of the Nixon quote in that Nixon is a paranoid nutter saying it, not coming from a rational place like Vance thought he just did.
Which part ?
Posting the entire speech only bolsters my view. For example
"[To accomplish goals].. I think are fundamentally dependent on going through a set of very hostile institutions - specifically the universities..."
I'm confused about your argument. I don't consider it a smoking gun just a concise example of what Vance and MAGA Republicans belive. There's no context confusion, it's on video, and it being dull only shows how comfortable he is exposing insane views.