Any anyone who's worked in a school knows how quickly diseases spread there. Like, my district, and all the districts near us, routinely close down every February come flu season as kids come to school with it (or other diseases) and they spread like wildfire. We take 3 days off, give the school a deep cleaning and let the kids pass their infectious period or get to the doctor, and then come back. Diseases spread like wildfire in schools.

This paper[1] on a Lancet sister publication disagrees, on the basis of literature review, that in this specific case the schools are a major part of SARS-CoV-2 infections. Of course there are many caveats in the discussion section.

[1] https://www.thelancet.com/journals/lanchi/article/PIIS2352-4...

From your source: "Recent modelling studies of COVID-19 predict that school closures alone would prevent only 2–4% of deaths" a 2-4% reduction in death rate implies quite heavily that children would be a disease vector.

Yes, and worse if the teachers die, since they're obviously at a higher risk of being in that extra 2-4% of deaths if schools do not close.