I think you could probably put something optional in place, sure. Some teachers will be happy to teach over the summer (if they get paid...), some kids will be desperate to get back & see their mates, some parents will need childcare as they go back to work in a difficult employment market.
Summer day camps with a focus on say education in the morning, then sports in the afternoon, for those it would suit, sounds like a good idea. You would have to work around maintenance & deep cleaning if using school premises, but I'm sure it's feasible.
What you can't do is expect teachers to give up their holidays a) for nowt &/or b) without them 'buying in' to the idea & being happy to go along with it.
& the reason doesn't have very much to do with the (toothless) teaching unions or whatever, it's basic market forces.
Teachers in the U.K. are a dwindling resource. I taught in the U.K. for 15 years then buggered off to a fancy international school overseas for twice the money & half the workload. It's great. U.K. teacher retention is appalling...with good reason.
If the government decided to expect state school teachers to work over the summer, on anything other than a 'the option is there & quite well paid if you fancy it' basis, the already unsustainable level of buggering off - elsewhere or just out of the profession - would increase exponentially.
& then you'd be screwed for September, when you expect things to be heading in the direction of normal, & suddenly there aren't the qualified warm bodies in the classrooms to teach anyone's kids.