I guess my brain has been cooked by internet libertarianism but what's the basis of a C&D for proxying a website?

Is the legal basis just copyright of the contents?

Bigger "guns" diplomacy. They don't need any legal basis for a C&D. Just depends on whether you have enough money to fight it in court.

i dont know the legal basis used, or the validity. i just know that i wouldnt want to have to spend the time or money fighting twitter over it

Well yeah the copyright violation is pretty clear cut here.