This is available in the US, but you make the choice based on the facility, typically. You can have the more medicalized experience at a hospital, or you can do home birth and hire a doula to assist, or you can go to a birthing center, which is sort of an in-between but more intimate atmosphere than a hospital. Usually the birthing center will be close enough to a real hospital to intervene if something goes wrong during the birth.

Which of these options does medical insurance cover?

My wife has had kids in all these settings, except the home birth was through the same provider as the birthing center. All of them were covered by insurance, except the birthing center is out of network. There’s an in-network one the next city over.

(This is a perfectly ordinary HDHP/HSA plan through a large company that you’ve heard of.)