Sixth form college will be the answer. I think context is important here. For people that aren't aware, in old school money, a 7 is an A and a 6 is a B, with a bit of stretch at the boundaries (i.e a low 6 could be a high C). So we are talking about 16 year olds who have just been through an extremely challenging couple of years, messed around with GCSEs are happening, then they're not, then they are again but no mocks etc etc. So 16 year old boys being told low As and high Bs are not good enough. And at grammar schools and private sixth forms, they aren't really, which I think is a bit shit, especially post Covid.
My son was potentially in a similar position. He is at a private school in South London. There you have to get 6 x7s to get into sixth form and 7's in the subjects you will do for A Level. Local state schools require 5s. The sixth form college I applied to for him as a backup requires 6s in the subjects you will take for A level. You get what you pay/tutor for I guess!
My point is, don't feel too despondent, there will be people like my son who have just scraped into sixth form and want to go there to stay with their mates - so will shortly be giving up their sixth form college place. And the sixth form colleges have less ridiculous expectations as well - round here the top ones want 6s for the A level choices but some will accepts 5s.
Having said that, we knew this going in right? (Although my sons school changed the requirement for sixth form from 6s to 7s after he started there). There's a reason grammar schools are top of league tables. It's a harsh old system for the academically more average though, no two ways about it. And remember, your and my son might only be average with 6s and 7s in the private/grammar bubble, but compared to the national average and other children their age they have done very well. I hope they both know that, but sadly the system they are in doesn't value them as much as it should, in my opinion. And doesn't make provision for the many, many people who are pretty average/immature at 16 and then improve/smash it for A levels/Uni as they mature (myself included).
Hugs, this is horrible, I know. I was very much expecting to be in this position myself. Phone round the sixth forms. I'd be astounded if they didn't have a place for someone with his background and grades by September.