They may very well Google how to abandon a baby before giving birth and would therefore learn about the boxs’ existence. And could take the baby to their nearest one straight after birth. Most people don’t go along pregnant and happy then randomly abandon their baby an hour after birth - usually they’re already planning to get rid of the baby to hide the pregnant and birth.
of course rare cases of psychosis might cause a woman to do this but unlikely so soon after birth if they’ve had the proper prenatal care.
trouble is money can be spent on making services available to enable a woman to feel safe having a baby but no one can control personal circumstances- very young teens panicking and giving birth alone, abusive parents/family, abusive partners, mental health, drug addiction, homelessness. Lots of people just can’t cope with the idea of police/social services being involved in their life for one reason or another
from the safe haven website they appear to have a hotline to help women as well as helping baby.
https://www.shbb.org/#:~:text=Safe%20Haven%20Baby%20Boxes%20has,Safe%20Haven%20Baby%20Box%20locations.