There isn't a bias against them, not at all.
There is a conscious effort to make sure that the bias towards them doesn't inadvertently prevent less advantaged children from getting places at top universities.
If a university is the type that generally makes AAA offers then the simple fact is that a very significant number of these grades come from students at private schools. Although only 7% of children are in private schools, 17% those taking A levels are in private. Within that, about 48% of A level results are at grade A or better in private schools compared to about 23% in state schools (with variations according to whether the state is a grammar, a comprehensive, an academy etc) so once all the numbers are crunched, you would expect a top university to have around 30-40% of its students coming from the private sector with no bias at all either way.
What universities are trying to do is take into account that it's a lot harder to get top grades in some state schools. If you were to take two students of exactly equal intelligence and potential, and put one in a nurturing and studious private school with loads of extra support available, and the other in a hugely underfunded comp in a challenging area, the chances are that the 2nd will get a lower grade than the first. So universities will give contextual offers where they accept a B grade from a student who got that B grade in extremely challenging circumstances, where a student who didn't have those circumstances would still be required to achieve an A.
That's not bias, and any family withdrawing their children from a private school to send them to a state for sixth form to try to game this will not gain any advantage thereby.