Government Funding and the Free Speech Clause:

Occasionally I see people suggest that private universities are bound by the Free Speech Clause if they get government funding, in the form of various research grants, student loans, and the like. But that's not the law: The Free Speech Clause applies only to decisions by the government (including government-run institutions such as public universities). It generally does not apply when the government merely provides funds to a private institution, that then makes speech-restrictive decisions without command or pressure from the government. The Court squarely held this in Rendell-Baker v. Kohn (1982).

The government may by statute impose many conditions on the use of government funds -- that's what Congress did with Title VI and Title IX (which generally bar recipients of federal funds from discriminating based on race and sex) and with the drinking age. Congress thus probably could mandate that no university which receives federal funds may restrict student speech. (I say "probably" because there are some twists which I set aside for now.) But Congress hasn't enacted such a statute, and it is of course under no obligation to do so.