See South Dakota v. Dole, which holds (among other things) that federal funding cannot be granted contingently on a requirement that would otherwise violate the constitution or be "coercive". This would almost certainly apply to the states via incorporation. So while a state could choose to not fund universities at all, if it chooses to, it cannot favor certain universities based on a speech-test.

Then you are arguing that these surveys would abridge free speech?

If done with the intent "to encourage intellectual diversity." Yes. At the point where you have intellectuals being required to register their views with the government, you're veering heavily into red scare territory.

These surveys are compelled speech. Freedom of speech not only means you are free to speak out, but that you can not be compelled to speak.