Sure, but to the second point, the vaccines also don't materially reduce transmission
Everything that helps to reduce prevalence helps kids stay in the classroom.
Here in the UK there's a huge subpandemic among 5-11 year old schoolchildren, a group which are not vaccinated. I have a child of this age, off twice with covid in the last few months. I wonder if it'd be more effective to vaccinate them or to just do away with self isolation rules for them. The problem is the spillover into parents, who will be at higher risk due to age and possobly at much higher risk if clinically vulnerable.
On this one I find it hard to form a strong opinion.
While vaccines should be available to any 5-11 year old child whose parents want them, the idea that they help in keeping schools open doesn't seem at all logical.
It's less true for Omicron, but the study there looked at transmission in Danish households. Against Delta, vaccinated individuals were much less likely to get infected. Against Omicron, boosted individuals are less likely to get infected, and vaccinated are similar to unvaccinated.
"This article is a preprint and has not been peer-reviewed [what does this mean?]. It reports new medical research that has yet to be evaluated and so should not be used to guide clinical practice."
Consider all pre-prints complete bunk until proven otherwise. Anyone can publish whatever they want to support whatever conclusion they want in a pre-print.
It's an analysis of an interesting data set and the conclusion isn't unreasonable.