Students from "nonprivileged" backgrounds are irrelevant to this discussion. Between 2017 and 2021 Cambridge increased its annual intake of POLAR4 Q1 students from comprehensive schools by only 19 - as opposed to 135 from the highest socioeconomic quintile.
The justification that was given for discriminating in favour of state schools, regardless of markers of disadvantage, was that it would meet the approval of the media, politicians, and public. However, educationalists believed that it would make independent schools less popular and force their closure (the addition of VAT to school fees was proposed for the same reason). One highly visible effect has been that independent school pupils who move to state schools for sixth form are 30% more likely to be accepted.
Regarding differential offers, only 23 students were accepted by Cambridge with sub AAA A-level grades last year. Approximately 40,000 pupils achieve 3Astar-A grades or better, which gives Cambridge more than enough wriggle room to discriminate against any subgroup that it dislikes without needing to accept anyone with less than AAA.
As to the possible argument that a 3A student from a poorly performing state school has the same potential as a 3Astar student from a "top" independent school, we know that this is not the case. Cambridge failed to find "consistent relative overperformance in Cambridge examinations by any disadvantaged groups with matched A Level attainment, and therefore did not provide support for the idea that the potential of the disadvantaged groups was underestimated by their A Level attainment, or for differential offers". We can deduce from this that "privileged" state educated students are admitted with lower prior attainment, because their degree results are substantially worse.
In other words, as for medicine, you can select for criteria other than prior attainment, but you will be found out.