A birth might be a flat $4000 but what happens if the mother has high blood pressure and needs to be on a magnesium drip? Or the baby is in distress and there's an emergency caesarean? Are you going to call around for a better rate?
The problem with our medical system is insurance. Insurance companies are always playing games to avoid paying bills and they force medical providers to play games to get paid. Vast amounts of money are wasted in billing and claims departments trying to process or deny claims.
It's gotten to the point where an insurer can fine the insured for submitting a claim with a code that requires prior authorization after seeking care from a healthcare provider. I'm not sure how pricey transparency would help with that.