You’re the one making the point. Back it up.
If the evidence used is based on influenza for schools then surely it’s based on that for everything? Even if we go with just the 0.35 for secondary schools that’s 7 times more than the close contact services they’ve shut. And ignores universities and primary school. The evidence is in the numbers. The only thing that changed in September was schools and universities opening.
So you can look at it from two angles. In places where schools have been open for months, has it led to a sudden surge? Alternatively, have there been sudden surges in places where there has been no changes to school measures?
When schools opened in Japan, did they have a second wave? They've been open for months now, and no second wave. But they got cases down low, so maybe it's not a fair comparison. Sweden have consistently had a much higher level of community transmission, and yet no spike due to the kids in school either.
What about somewhere like Belgium, where they re-opened schools at the beginning of September, but throughout September had around 2,000 cases per day, steady as you like. No change to schools in October, but things jump off the charts to 6x those levels. Maybe it's just a delayed effect? Instead we could look at when the peaks preceded the schools re-opening. What about in the US, state by state? Florida, Texas, Arizona had their worst period when schools were closed, and while they're on the rise again now, they're nowhere near the same level. In South Africa, they re-opened all school levels at the end of August, after getting over their first peak in mid August, and as we enter November there has been no rise in two months - 1,500 to 2,000 cases per day.
We know that the thing that you deem inexplicable - a sudden explosion in cases - has happened before schools re-opened and long after schools re-opened. We know it hasn't happened at all in cases where schools have been open for months. In other words, it's entirely possible that the explosion in cases happened irrespective of schools re-opened. Otherwise you would have seen an explosion in cases in Sweden, after many months, or you wouldn't have seen an explosion of cases in the US southern states while they were off school as normal.
All you've picked up on is a correlation. Many more people get murdered when ice cream sales go up. One didn't cause the other but they were both driven by the same underlying factor. Schools re-opened as we progressively relaxed all other aspects of life. Lots of things have changed since September, most importantly, people's attitudes. We opened schools when we felt comfortable with things, and when we felt comfortable with things, we spread the virus in normal social settings. Schools play a role in transmission but they aren't a dominant factor, and most of the role they play has nothing to do with kids passing it to each other in worryingly high numbers.