If herd immunity is not the goal then why not lockdown sooner so more ventilators and beds can be available through buying more and then going into the summer. Surveys of children to divide into those who can easily work from home, those who would need some help and those who have parents who are deemed essential workers. Set work remotely for all children and they either work at home or in a childcare setting.
As I understand it one of the main problems with holidays in school time is that some children miss one week and another child misses a different week. This will be the same with covid 19 as they will all be ill at different times. By having three months work (obviously not all set at the same time) they can catch up if they are ill and then all catch up together when schools open. Yr11 and yr13 come in just for exams. As all children have had same disruption and as long as a similar percentage get certain grades each year we may just need to accept that education is an area which has some collateral damage for a few years.