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.