We have sensors and ran HEPA filters all through the Covid period and for two years beyond as we were part of a ventilation trial. Turns out, even with the filters running and the windows all wide open, you simply cannot get the CO2 levels under 800ppm when there's 30 kids and an adult or two all sharing a tiny space. If you want ventilation to be the solution, it would take more than just windows and a few filters, you'd have to redesign and rebuild classrooms/schools.
As an aside regarding cost, schools don't pay for supply teachers anymore. They tend to cover internally by using TAs because they can't afford it. They'll only hire supply for longer periods because the insurance kicks in and covers the cost.
Having children off when they have colds/coughs would help with illness in school but parents are unlikely to support it considering they already try to sneak children into school when they are more actively sick (vomitting, rashes, etc.) because they can't take the time off. To change that, you'd need a change in employment law, not in schools.
As for masks, we couldn't even get the general public consistently wearing them during a global pandemic when it was still of unknown danger and risk. There is no way parents would support children being masked in school as a regular thing.
As a teacher that gets sick every single Christmas holiday, I agree with the point that something needs to be done. But your solutions are oversimplified and would be unlikely to be effective.