Maybe the issue lies more in your first statement that children need to go somewhere. That's a fairly modern notion. If we've created a society where we rely on strangers to care for our kids, then maybe that's something that needs to be looked rather than creating additional covid hotspots?

While we're doing that, maybe a compromise with A/B school days to limit student numbers, with online participation for whichever group is at home. Schools also need to create strict rules as far as cafeterias, masks, etc... With clear consequences for violations.

The school I am at has dividers in the cafeteria and staggered lunches with no talking while eating. Masks at all times. Temperature checks at the doors and such. This is countrywide (not US) and there has been no spread in schools.