Statistics apparently suggest that those who went to private schools are more likely to vote Labour.
I don't disagree that accepting ideology, whether gender or whatever, is not solely an Oxbridge problem. However I wonder whether top Universities both in the UK and the US will suffering relatively more reputational damage if they accept things without being open to debate and alternative views. Thinking is what they are supposed to do, so cancel culture becomes very damaging.
London Universities seem to have done better. Perhaps because UCL was formed as a non religious alternative to Oxbridge. Perhaps because of the high numbers of overseas students and the "vocational" nature of so many of Imperial and LSEs degrees. Not perfect, but on gender good female academics seem to have been allowed a voice.
I know a couple of young people who have struggled because of they were open about their support of the Conservative party, even though it was the Government of the day. In the peak "Oh Jeremy Corbyn" days it was pretty grim to be a Tory at Oxbridge, even though the young person was very bright and personable and happy to debate. More recently a second was in despair when the only non-Marxist academic in his University in London department (not politics!) left and he essentially started treating his degree as a wfh qualification that he cynically progressed along with an interesting and career enhancing part time job. (He did perfectly fine!)