Simple. If the college accepts federal funds of any kind they become actors of the state and are bound by the Bill of Rights and the Constitution.
Greetings:
I mean it's OK to opine that you wish this is the way things worked. I think your opinion would be the wrong one, but no matter. The reality is that this is not the way things are. If a private university is taking money from Uncle Sugar understand that they are doing so VOLUNTARILY, and most assuredly with a full understanding the terms attached. The government may place all sorts of rules based on your acceptance of that money, and the worst that will happen is, "Hey university, cut that out or we will cut your funding." Keep in mind, we are talking about a private university here. Coe is a private university.
In these cases, you need to understand that private universities are not required to protect speech the way public ones are. As far as free speech when giving research grants, for example, the way it typically works is that the government will say, "We'll give you this grant, but you have to design your own institutional rules for how you're going to protect free speech." Absent is any methodology for how that is monitored. Absent is any approval process for the institutional rules laid out. Absent is any concept of what does and does not violate free speech. You just have to say you're going to do it, get your research grants, and find bright PhD prospects to fund.
In a situation like this, if a violation of free speech is suspected, the government will say, "OK. You're not getting any more money." The university will usually back down because research grants are fairly coveted and in many cases lucrative. I hope you understand that backing down because of the fear of losing money doesn't mean a violation of free speech was committed. It simply means the university doesn't wanna lose that cash. You understand that, yes?
"Bound by the Bill of Rights and Constitution" merely for accepting the federal funds? Nope, not the way it works. Not even a smidgeon. If this is your position, surely you can back it up with court cases which have been adjudicated against the university using this logic.