Yes, Belgium have had an increase in cases. Your hypothesis is that that sudden rise in cases is best explained by schools re-opening. Nothing else changed, so how could it just spike like that? Apply that argument to the southern states in the US, which had a sudden rise when kids were all off. Apply that argument to South Africa, which had a very similar spike at the same time, and 2 months after re-opening schools, has kept things pretty steady. In other words, is it possible to have a sudden spike without any change to school procedures? Absolutely. Alternatively, is it possible to keep case levels steady even after re-opening schools? Sweden, Japan, South Africa two months on...many other countries. Yeah, it is. It is not an irrelevant factor, it plays a role, but your suggestion is it is a dominant factor, the best explanation of this change. It just doesn't fit when you apply it outside this context you first applied it to, the causal explanation falls apart.
You don't need a change in a single factor to explain the sudden spike in cases. It happens independently of schools. There's a reason why the evidence you cited said "remains unclear how infectious children may be" and "unclear how much schools may contribute to community transmission ". They don't have an agenda, they closed schools anyway out of an abundance of caution and convenient timing. It's just the pattern you believe is so obvious is apparently undetectable to them, the people with all of the data.