Sounds like it can't go wrong, sovcit.

This should work. Healthcare should be a free, no-questions-asked (within reason, obvio) service available to anyone at any time, regardless of who/ what they are.

This is the birth of a human child. Healthcare tends to want to protect children, even if the parents do not.
- Which is highly situational and requires the judgment of those trained to make those judgments (aka. child social workers or family therapists), not a blanket policy of requiring invasive and non-medical information from those simply seeking medical care.
  
  In an obviously abusive situation, the abuse is clear enough that it can be addressed in the time it takes to give birth and examine the tiny human. In less obvious abuse cases, the parents can almost certainly make it through the entire visit without raising suspicion, regardless of the information provided (though it might make them easier to identify if the abuse is discovered later). There should be an observation period where nobody gets to go anywhere, but again, that should be limited to only what is medical in nature.
I mean, they would deliver the baby. That's how healthcare generally does work - even in the US - for emergency situations. Only issue is, if you won't answer any questions, they'll probably keep the baby, for a number of arguably good reasons. Human trafficking, for starters.

When you finally sign the forms you need to sign to convince the hospital to let you have the baby, then you will get hit with the horrifying dystopian medical debt. Or you will pay (almost) nothing if you live in any sane society.
- If you pull a screaming squishy blob out of the vagina of an equally screaming, slightly less squishy blob, I think it's pretty obvious who the squishy blob belongs to. If there is abuse of some kind, there is almost certainly other signs (defensive behavior, unexplained injuries, uncomfortable relationship dynamics, etc) that beg medical questions and don't necessarily rely on getting information like socioeconomic status or personal beliefs.
Sovcits have actually showed up at hospitals before and CAS has gotten involved. One was a sovcit couple who were trying to deliver baby themselves out in the woods in a state park in Kentucky I think; labour stalled after 2 days so they finally came in, and during the admission told the hospital staff they planned to move back into the tent with baby when born, which was during the winter, and refused to get baby a birth certificate or give him a name, so CAS took him and their rights were eventually terminated and baby adopted out. Dad later got arrested at a library for threatening people, and they still have Facebook page demanding his return several years later.

There's also an OB nurse who has a great Tiktok about obstetrical care and birth, and she says she has helped deliver three sovcit moms after they finally gave up trying to deliver unassisted and came in, and two of the babies didn't make it.
- Those are situations where intervention and questions are very much appropriate, but only because there were other factors at play that were medically relevant. Medical care shouldn't be contingent on the simple description of Sovcit. At most, it might raise a couple eyebrows, but there is likely to be far more important and empirical signs of abuse that are less prone to bias and prejudice.
Since they don't believe in government or any kind of public assist, they should either pay cash or trust their god.
- Or, ya know, be given the same basic respect and rights as any other human being. Personal belief lands under the 'don't ask' category, and should never be used as a factor in medical care unless the patient themselves makes a belief-based request.
It pretty much is that way in the rest of the "free world".
- Ok, try going in to a hospital in Europe and not give them any information, not including even your name or whether you're allergic to anything.

To add a note given our own experience, labor and delivery depts tend to be on pretty heavy security around here. We're talking wrist bands for everyone, locked and monitored entries, etc., because people kidnap babies (usually fathers w/o custody or have ill intent). Plus, mistaken indenties happen so you get a matching band with your kid.

They also almost always have post-labor care mandatory since you're practically wheelchair bound after such a traumatic experience. No just walking out of there, you physically can't.

All that is to say, yes, you have to give your name and information and no, you're not just getting a baby to go. They don't exactly want a rando entering and leaving with someone's child, even by accident. You do have some rights though, at least around here, since for some reason we still respect even the craziest of religious rationals (like no meds at all).

In the UK it's two tags. Baby's like to riggle out. Of them.

They definitely don't make you fill out any forms, put trackers on you and the baby, and constantly check in over days. I'm sure it's super easy.

This reminds me of what a former student is going through. Her boyfriend's parents did home births for all the kids. No birth certificates, no social security cards, no records. Parents just decided last month to up and leave. The 18 year old just got dumped with his siblings, a farm, and no safety net. So the girlfriend and some other friends that just graduated have been working together to care for these kids and run the farm while they work with DHS to figure out how to go forward.

That's sovcit behaviour. Call CAS. Not having identity for your kids will get them taken away.
- Parents are already gone.
- I don't think the parents are coming back... They gathered up all the cell phones and electronics and destroyed them before they left. The kids are receiving help from the proper agencies.
That's completely psychotic

When we had our kids—in a small country hospital that my mother in law worked for, mind you—we nevertheless had to prove our identity multiple times, and I (the father) was put through extra identity verification—including the go-ahead from my wife—because there are so many liability issues around making sure that the right parents get the right kids. One time we were literally the only couple—and our baby the only baby—on the entire maternity ward and we still had to go through the wristband/umbilical clamp verification process, which included our names.

Sovcit's about to have a really fun time at a hospital.