there's still the practical issue of how in the hell to get health insurance to cover costs for that stuff.

As you say, hopefully in a few years the health care landscape will have changed for the better. But certainly it can be done - a single-mom friend of mine had to deal with this (she identified a midwife practice with a payment plan and started making payments before the baby arrived). Even with that, probably still cheaper than hiring an immigration lawyer, all told. :P