No hospital would give out that information. Doing so would be a violation of HIPAA, as well as State privacy rules.
If you want to test this, if you have a child or other family member born at a certain hospital, call that hospital and ask if they were born there and see what they say.
You’re far too clever for me.
This hospital verification thing seems more to the point than this seemingly invented distinction between “born in the U.S” and “natural-born citizen”.