My wife and I just welcomed our new baby girl into this world just 4 weeks ago. Speaking as someone that just went through the whole ordeal (other than actually birthing the child) it would be pretty hard to nab a baby in today's modern society, at least in a modern U.S. hospital (it may be the same for other countries, not trying to be xenophobic. Just saying my experience relates to the U.S. and OP is from the U.S. as well).

Pretty much right after our daughter was born they put a tag/bracelet around one of her ankles that is basically like baby lo-jack. If the bracelet is tampered with, falls off, or leaves a certain perimeter (basically the labor & delivery area) it sets off an alarm and the entire floor goes on lock down. Elevators stop working and all exits are locked (although now that I think about it what would happen in the case of a fire?). There were 3 or 4 false alarms while we were there so the tags are pretty sensitive, but better safe than sorry right?

Nurses are also trained to look out for that kind of stuff. For example, at our hospital if you wanted to walk around outside of your room (like in the hallways to stretch your feet) with your baby you had to leave the baby in the little bassinet/crib with wheels and push that around. You weren't supposed to walk around with the baby in your arms, the nurses didn't even do that. If they saw someone just walking around with a baby then that's a huge red flag.

Now our experience likely wouldn't be just like what OP goes through, but her hospital probably has similar safeguards in place. Clearly the scammer has some serious mental health issues.