Government loves anything that assists its people-tracking machinery. It will never ever ever discourage pregnant women from doing something that would make reliable Enumeration At Birth more difficult.
As fun as it sounds to go back to 25-50% infant mortality, I would prefer we keep this in hospitals and do cesarian sections and other interventions when necessaru.
It's also very expensive to be underdeveloped. Simple things go from given to impossible problems. Things we take for granted, like sanitation, because insurmountable obstacles. Just ask any developing countries.
Less infant mortality is good, actually, because I like living in a developed country. I don't want to burn our tax payer dollars on burning baby corpses. I don't want to burn tax payer dollars on a stupidly volatile economy because everyone is dying all the time.
I see your drill down to fundamental issues and I raise you:
Pregnancy is not a sickness. Hospitals are for sick people.
If we reserved hospital birth for women and children with actual medical problems we would allocate resources much more judiciously.
An added benefit: not exposing otherwise healthy (and capable) women to an almost universally disempowering and disenfranchising birth-industrial-complex that seems designed to engender fear, self-doubt and pathological outcomes.