The choirs are all fine though? I don't know about Oxford, but at Cambridge, it's only the Kings College choir that is all male (with thd boy choristers coming from the nearby school set up for this purpose). All the other main college choirs are mixed. They only have to find a handful of each voice type, out of hundreds in a college including postgrads. Why would they need to recruit from the local area?
Also, it simply doesn't happen that boy choristers are somehow fast-tracked into 'top public schools'. Where do people get this notion from? It's not 1850. These schools are, first and foremost, academically selective. In any case, being a boy chorister doesn't mean you will still have a decent singing voice later - depends what happens when the voice breaks.
Yes, some schools offer better choir training than others and many students will never have had the chance to take singing lessons or be in a choir. But they do have to be able to sight-read quite well, certainly for a choral scholarship, so it's not as if the choirs are suddenly being filled with anyone who fancies a bit of a sing-along.
Thet have to send in a video of them singing a piece from a set list. This is AFTER a conditional offer has been made. Then, it's the intercollegiate choral trials. They sing one piece unaccompanied. One piece sight reading. Then there are other tests where they have to differentiate between notes / chords, sing notes back etc. it's not really the kind of thing students would put themselves up for if they didn't have at least Grade 8 singing and some choir / solo experience.