I don't see any major problems with this
I see no issue at all with the only cold lunches being provided aspect of it. Hot lunches at schools are nearly always very unhealthy and no one NEEDS a hot lunch to make it through the day. With the sack lunches available to purchase there is still a way to provide free or reduced lunch to those who need it and still food around for kids who forget their lunch, etc.
I don't think kids will be socially stinted from eating in their own classrooms. They still get 30 minutes out on the playground with all of the other kids.
They will probably end up with More time to eat and to play as much less time will be spent getting everyone down to the cafeteria, through the lunch lines, etc.
Overall I think it is a great idea of how to relatively painlessly boost the budget.