In the US, you'll be billed the maximum out of pocket for your health insurance whether your little one is in the ICU for months or whether it was a easy delivery though, so at least birth is a predictable medical expense.
I asked one of their counsellors once if they ever have any ultra wealthy people who don’t have insurance and also don’t qualify for any assistance. (This was at a children’s hospital with a Level IV NICU.) She said she was unaware of that ever happening, other than very wealthy foreigners who would prepare in advance, arrange payment in advance, fly in, and have a special procedure done.
Overall, in many states, it is logistically impossible to have an unaffordable bill and also not qualify for assistance. The worst situation actually is the person who has insurance but has high deductibles and copays.
If you do not have insurance, you tell them you don’t. They’ll give you a bill lower than what you’d pay as deductible if you had insurance. Or you just don’t pay…
This raises insurance premiums and reduces the quality of healthcare in a dystopian AF feedback loop.