But it's hard to know if the rise in September was due to schools, or was it due to 'Eat out to help out', pubs reopening (and let's be honest, a lot of pubs in London at least did not stick to rules about 'family groups'), universities reopening etc. Or a combination of all of the above.
Eat out to help out and pubs reopening started a lot earlier than schools, though. I do agree that lots of pubs got more lax with the rules over time.
I also agree that universities reopening had an impact.
But there are lots of break downs in cases by age that show a sustained rise in cases in the secondary school age group (11-17) from September onwards, increasing especially after October half term, when cases in other age groups were falling due to lockdown.
I don't think that many secondary school students are down the pub or regularly eating sit down meals! Obviously, they may be socialising elsewhere, but schools being open has an impact on this (i.e. parents allowing their children to go around someone else's house because they are in school all day).
At the end of last term, there was also a clear rise in the primary school age group, as well and increasing numbers of primary age children were off school.
I think there is clear evidence that schools being open with no mitigation measures do contribute to virus spread- they are not the only cause of it, but they are a problem.