This is a great idea, and I'm really curious why we don't have something like this, just mail a free mask and a free test to every address once a week, with a form to request more for larger households, or to request that they stop.
https://www.gov.uk/order-coronavirus-rapid-lateral-flow-test...
It would cost ~50 euro per family per month, which is quite expensive, but possibly if they use some sort of model to give tests to people with higher probability of getting sick (e.g. if your child was in a school that got N positive tests recently) maybe it can be reduced to 5 euro per family per month.
Somehow I feel this is a missed opportunity to get things a bit better, I dont think people will infect other people on purpose.