Do you have a link?
I have been an admissions tutor for a very highly sought after course at a prestigious university for over a decade as has DH (I appreciate that having said this I will now inevitably litter my posts with spelling errors etc!) I would be stunned if there were a preference in the crude way you suggest. It would be absurd. E.g.
Child A - 2 degree-educated parents with professional careers. The parents purchase an expensive house in the catchment of a well-regarded state school. One-parent goes part time to support child after school They have enough money to spend on private tutoring, music lessons, 'educational' trips etc. Child state educated.
Child B - started in the same way as child A but was horribly bullied at school, so the parents decide to go full-time, cut the extras and send the child to the private school with the good pastoral reputation.
Child C - Refugee family, very low household income, parents speak very little English. Wins scholarship to private school and this is topped up with a large bursary. Works exceptionally hard but has very little academic or financial support from home
It would be ridiculous to prefer Child A over B or C. There are loads of 'child A' applicants to our course, I imagine quite a few Bs and I have seen several Cs (or similar).
Rather than a simplistic state v private we have increasingly detailed data about the school background and the performance of that child against the performance of others at that school. We also have data that is designed to highlight other forms of disadvantage e.g. on the basis of postcode or time spent in care. It is certainly the case that a child who has not performed well by the standard of their school will find it difficult to obtain a place on our course, even if a child with the same grades from a more difficult school does get a place. We do not make simplistic assumptions from the sector the school is in.
What is difficult is measuring the home advantage a 'Child A' has against another child from the same school without the home support and advantages. A preference for state over private will often simply work to advantage 'child A' and do very little to address the disadvantage that pupils with a more difficult home environment face.