In America, you basically need insurance to act as a larger stronger party in the negotiation of prices with the hospital on your behalf. Without the bargaining power of the insurance company the prices you'd pay can be significantly inflated. So paying for the insurance is the slightly lesser of two evils. Supposedly. From your question it seems perhaps you live somewhere with a saner system in place. I'm envious.

Birth injuries are insured by the ob/gyn, which is part of the very high cost of delivering a baby in the US (because such injuries often lead to a lot of care).