This may just be conjecture on my part but I think the dominance of oxbridge may in part also be due to self-selection out of the process by many students that went to lower ranked / non-RG institutions. I myself went non-RG for undergraduate and many undergraduate friends just won’t even apply to more prestigious law firms as they have decided they don’t stand a chance. I also think A level requirements play a role here, those that meet the A level grades of AAB - ABB require by management firms are more likely to have gone to oxbridge/RG universities.
I do agree that better universities get better access to law firms - my undergraduate university law fair was not visited by MC/US/SC/city law firms whereas they almost all attended the london RG I did my masters at and held exclusive events for that university as well.