As far as student unions go, the key point is the No Platform policy, which was adopted by the NUS in the early 1970s. Any Labour MP who has come up through student union politics, which is most of them, takes No Platform to be axiomatic. The policy has been around since before many of them were born.
The policy was originally designed to deal with the National Front, who actually were trying to organise on campus at the time. But today the NF is irrelevant and its offshoot the BNP almost equally irrelevant. Neither has even tried to organise on campus for decades.
However, in the world of student union politics the demand for No Platform vastly exceeds the supply of fascists, so they just expand the categories of wrongthinkers who it can be applied to. Julie Bindel has been officially No Platformed by the national NUS since 2011.
One might set this alongside some student unions' periodic attempts to derecognise Jewish societies because something something Zionist settler colonialism.
Interestingly enough, the concept of No Platform was invented by the now defunct International Marxist Group, who at the time were noisily supportive of the IRA.
So as far as student politics goes, opposition to the very concept of free speech is not just something you find among Trotskyites. Even quite moderate Labour people work on the basis of a Schmittian friend/enemy distinction. That goes a long way to explain their strong opposition to the Act.
Of course it's also very likely that Phillipson has been lobbied by VCs, but that will just have encouraged her to do something she wanted to do anyway - and that Labour didn't mention anywhere in their manifesto.