You can actually do this. Get your treatment, receive your bill (which you must pay), and then ask your insurance company to pay you. Eventually you get a check in the mail from your insurance provider.

I've done it for a few childbirths with midwives and I've done it many times for dentists.

I think this used to be standard, and may relate to that whole annoyance about you being responsible for whatever charges aren't covered. When you give your insurance info to the medical provider, you're not changing who ultimately pays. The medical provider is simply offering two things: they will handle the paperwork and they will loan you the cost of the treatment until the insurance claim is processed.