the Brief Symptom Inventory, a short version of the SCL-90, is in wide use for screening adults. it takes 8-10 minutes to fill out. it is reasonably valid for adults. i don't know if there is a version for kids yet, but you could probably do a ten minute interview version using minimally trained technicians.
the technical part is the easy part. the only thing to stop this would be popular opposition.
Do you see that as a problem?
What do you propose to do with those citizens who do not want the government probing into their minds?
Does "privacy" have no meaning other than "the right to abort"?