I changed career to be a teacher a few years ago so I do understand what it's like to have a more usual work pattern as a parent. Industries vary but I knew very few people who couldn't vary their hours enough to be there for a special assembly or exam. When it came to school holidays, however, DH and I split our AL so we could cover them and never took time off together. There was lots of ferrying and timetables.
When I trained as a teacher, I explained to the DC that I would be there for the holidays, but that I wouldn't be able to do the events and special days. During term time, they know I need to work most evenings and at least half a day every weekend. It's the first day of the summer holidays today and I've spent half a day in school sorting out my classroom. I'll go in at least one more day. That really is the baseline just to provide the level of teaching needed.
So 190 days of teaching, plus 5 days of Inset, plus around 5 days during school holiday time. I know lots of jobs require working outside of normal hours, but if I only count the Saturdays, that adds up to another 20 days or 4 weeks. So it works out around 8 weeks' holiday. I absolutely, truly, honestly appreciate and value that holiday and understand that it is a perk of being a teacher.
Here's the other thing: I see your children exhausted by this point in the year. Yes, there is some slippage during the summer which we account for in our planning. But they also come back refreshed and ready to learn (whatever they've been doing in the holidays).
In terms of when the holidays are taken, I think the summer break needs to start earlier, and this may necessitate reorganising the weeks. My suggestion would be to start at the beginning of July and return in mid-August - but that leaves a long autumn term. The May half term week could perhaps be redeployed to the October half term? There would still be a long weekend in May and the one in August, so those short breaks would ease the change.
It's interesting that as pp have said, many countries with good outcomes start school later and have longer holidays. I think they are also set up better for affordable childcare so perhaps there is less resistance. Additionally, you would expect private schools to have shorter holidays if there were such a danger of knowledge loss, but their holidays are longer too.