Furthermore, the people most likely to kill and abuse their partners and children are men, and I don’t see any suggestions that all men need a psychological screening.

It should be noted that women are much more likely to kill children in their first year of life. With the seriousness of postpartum mental health issues maybe we really shouldn't be leaving mothers alone with their children? I mean this in multiple ways, including that recent parents should have far more support including visits by a nurse or midwife, social services, etc. to make sure parents know what to do and have what they need to do it. Our society leaves people to fend for themselves far too often.