the BBC can't be very biased because all sides accuse it of being biased against them
I don’t think that’s true. For example you won’t find many Remainers claiming that the BBC was pro-Leave. Nor would you find many opposing Scottish independence saying the BBC was pro Scottish independence. It clearly does “take sides”.
The traditional left-right axis doesn’t work as a model here. The BBC’s bias is towards the opinions and interests of upper-middle-class Londoners, some of which would be traditionally be considered left and some of which would be right.