They are not supposed to. But it could happen. Worse what happens if it is a heart attack, stroke or vehicle accident, and you aren't able to buy insurance immediately? You would be liable for all expenses until you purchased it.
What if it’s not an emergency, as defined by EMTALA, but something like cancer? A hospital does have to take you in and stabilize you if you are having a heart attack, but not if you find some hard lump, or persistent low-grade bleeding.
Let that progress until you’ve got a ruptured bowel, or seizures because of the brain tumors - yeah, then they have to stabilize you. But not until then.
I see a new underground business cropping up. You pay an insurance agent $100 a month to have everything that’s needed to sign you up on hold. If you have a heart attack, first call the ambulance. Then call your “holding agent”. You could have an “In Case of Emergency” card attached to your driver’s license instructing the paramedic to call him for those times you’re not able to make a call. The “Holding Agent” will have instructions to call your family.