That's not necessarily true. There's a whole body of law surrounding when the federal government can and cannot engage in quid pro quo withholding a federal funds. Much of it was litigated with respect to the withholding of highway funds in exchange for compliance with Federal blood alcohol limit standards and the like.
Much of it was re litigated with respect to title IX educational compliance.
I can't speak to specifics, but my understanding is that there are limitations on the withholding of funding and it must be directly tied what the funds are actually being used for.
As you say, Columbia is a private entity, so that is a major distinction. However, there's also a body of law around selective punishment for speech. An example of this came up when the White House selectively punished the AP because it did not like the content of its speech.