The equivalent would be that if any employee of any company uses their cell phone while at work to post something on twitter, the company they work for are responsible for the content of that speech. Even if they are on lunch break, if they do it while in a chair owned by the company, then the company is responsible.
Or a cable installer is out on the job, they decide to go home and kill their wife. The cable company is now responsible for that murder b/c he was on the clock at the time and used the van to drive home?
Currently, companies are not liable for those things. They are liable when an employee causes harm as a product of them performing their work duties, which includes harm caused by incompetent execution of those duties. Companies can be expected to ensure their employees are competent enough not to cause accidental harm to others. They cannot and are not expected to control every personal decision an employee makes to do something outside the scope of their job. The exception is if they ignored obvious signals that the employee was dangerous or mentally unstable.
So, it seems reasonable to you that every University hire thousands of additional full time lawyers to screen every single word ever uttered or typed by any employee, whether on campus, off campus but in official capacity, off campus in an unofficial capacity but using a laptop paid for by grants, etc..
How is this an affront to free speech? Libel/slander is not protected speech.
If Universities could protect themselves from such an absurd precedent (which would require billions of additional dollars), it would require them censoring speech prior to it ever being found to be libel or slander. Any possibility of it being so would have to be censored. And since about 1% of things that could possibly be found to be slander by a jury are actual slander, that means 99% of what is censored by the University would be legal speech.