If you want more mitigation measures in schools though how long do you want them for?
Primary school aged children may never be vaccinated (with possible exception for CV children at some point) because on balance the vaccine may pose more risk than covid in that age range. Same may be true of secondary though perhaps more likely that they will end up deciding that there is some benefit in vaccinating at that age.
Covid may therefore spread at a significant rate in at least primary schools indefinitely if, as experts are saying, it has become endemic.
Despite mitigation measures (including the extreme measure of closing schools for much of this year and the quite extreme measure of isolating close contacts) around 25% of school children have caught covid in the last year. It seems plausible that that rate of infection will continue in schools even if current measures are retained or perhaps it will be a bit lower due to some immunity from adults and some immunity in children who have already had it but you might still get significant spread amongst children.
So I would think that if you have a child starting primary school it is highly likely that they will catch covid at some point before they are at an age where they can be vaccinated even with very disruptive and extreme mitigation measures in place (like closing schools and multiple long isolation periods and cancelling most extra-curricular activities etc.). Covid may just be an illness which children tend to get at some point during childhood.
So if a child is going to catch it during their time at school anyway and this is judged to be lower risk than vaccination at least until they reach a particular age, what is the benefit in mitigation measures which slow the rate of spread but don't stop it. You are basically putting these measures in place to reduce the risk of getting covid per year from (say) 50% to 25%.
As a medium term measure while adults get vaccinated I can understand the argument (though I think the harm done to children in the process far outweighs the benefit to anyone), but are people saying that they want permanent measures to slow the rate of covid spread in schools if it is decided that it is not appropriate to vaccinate children i.e. children in masks and testing for covid regularly for years, missing c. 20 school days a year due to self-isolation, no mixing between school years, limited school events and trips, no parents in the school etc.
I'd personally prefer that my child caught covid at school than have that sort of school experience as the latter sounds much more damaging.