There's also the issue that the fatality rate for the <19 year age group is very low, about 0.003%(1), and even lower for the slightly older school aged crowd (approximately 0.001%(2), 1/10,080 of the 85+ demographic). It'll take time and large studies before we can actually prove that a vaccine is safer than COVID-19 for that age group. Note how a previous H1N1 vaccine turned out to cause serious narcolepsy in a minority of people: https://www.cdc.gov/vaccinesafety/concerns/history/narcoleps... There's also the problem that you need enough people in the test to be exposed to COVID-19 naturally: vaccines can in some circumstances make infections worse - an issue that has made developing the closely related SARS vaccine hard - and you need large studies to detect that in cases like young people where the disease isn't very deadly to begin with.
A safer option would be to provide vaccinations for adults working in schools, starting with those most at risk, and working down to least at risk as safety evidence builds up.
1) https://www.cdc.gov/coronavirus/2019-ncov/hcp/planning-scena... 2) https://www.cdc.gov/coronavirus/2019-ncov/covid-data/investi...
Unfortunately this will be quite some time away still. None of the vaccines currently in phase 3 trials are targeting approval for use in children.