Summer breaks are useful for parents of means who can put their kids into other activities or parents with commercial activities with which the kids can help. For most everyone else it is beneficial to have year round 8-5 school, not because kids get more educated but because it is highly effective babysitting.
A case can be made for a trimester year round school situation in which students are required to attend two per year. Two in a row can be missed with permission, and missing three is equivalent to dropping out of school. This could be beneficial in many ways, but would be complex.
A big problem with the proposal is simply costs. Right now there is a lot of stuff that happens in the summer. Some of it is paid by school budgets, some of it is paid by external grants, some of it is paid by the teacher. The fantasy is that we can increase the school year with no significant costs. This is not true. Over the past 20 years teacher pay has gone up considerable, and a lot of that cost has to do with simple additional time the teacher is required to work. This is the same in any situation. If you are paid hourly, i.e. required to be at work at 7 and work until 3, if they ask to work until 4 every day then that should come with additional pay. This is what has happened with teacher, the additional few weeks and time per day has increased pay about 15%. If we go year round the pay will increase another 20% at least.
There will be other costs. Training will not happen during the off time, so staff will have to hired to cover classes. The argument cannot be made that personal business can be handled during breaks, so teacher will have to take days off during the year, a practice that many teachers now try to avoid. This again will require additional staff hire. To give you an idea of this additional cost, say a school has 60 teachers, which is a small school. Two weeks of training and two weeks of personal time is 20 days. For all teachers that is 1200 person-days, let round it up to 10,000 person-hours. At minimum wage, rounding up again, is $80,000 per school in class coverage costs. Not all of this is new costs, but it is significant. Add a half million for additional pay, weekend costs to maintain the school that is now down over break, and one is looking at a cost per student going from around $7,000 to around $8,000.