Despite being a British citizen, I don't understand the university model all that well, but one argument I see being banded about over the creeping levels of co

Despite being a British citizen, I don't understand the university model all that well, but one argument I see being banded about over the creeping levels of control over UK universities is why the more elite-level universities haven't opted to go private. Sure, there was a legitimate argument a while ago with no/low tuition fees. The government was footing the bill. Now that tuition fees have risen, and seem to want to continue to rise, what benefit is there for the likes of Oxford, Cambridge, UCL, and co to remain public? Someone I used to work with attended the University of Buckingham, one of the few universities in the UK that is private, and he said that while it was no Oxbridge, the flexibility of graduating in two years made it completely worth it. Ironically, a lot of pressure is put on the ex-polytechnics and middling universities in the UK to justify their high costs, low quality of content, and average employment figures, with the government often pointing towards the need for flexibility (i.e. two year degrees to save money). If the government want to get their claws in, I'm surprised that those with the freedom to go private haven't decided to, or at least as a way to one-up their competitors.