To be precise, not really a doctor: the pre-natal care program is generally handled by dedicated professionals whose training is that of a nurse plus some specific to pregnancy. The job title translates "midwife". The system is called "neuvola", literally "place of advice". It's actually a word devised for this very purpose.
This system was established in 1940's and that's also when the baby packages were first given out.
But expecting mothers don't really see a doctor/physician during the pre-natal period unless they are actually ill somehow. Pregnancy is not an illness.
You would be surprised at how many babies are born in western countries without prenatal care, simply because the mothers-to-be didn't know that it was something that was available and needed. Encouraging new moms to visit the doctor is a fantastic thing. Besides check-ups, it encourages vaccinations and can help with early identification of congenital problems.
The box is used for bait. New parents want the box, it's free and full of great stuff. To get the box, they need to visit a physician. As others pointed out, a single visit to a physician can help to identify issues with not only baby but also mom.
Pre and Post-natal care are two of the largest factors in lower infant mortality. By using the box to get Moms in to a doctor's office at least once, many difficulties can be found and overcome.
