I carry my health insurance card in my wallet, but every time I've been taken to the hospital (I've had a few ambulance rides the past few years) I didn't need it when I got to the hospital. I gave them them my name, DOB, and company and they just looked it up in the computer.
The hospital I got to for all my surgeries and testing has the information on file, and it hasn't changed in years. I get coverage through my employer, and will keep the same coverage after I can retire (in 2 years, 16 days). If anything changes on that front, I log into the hospital's portal to update it.
As far as emergency care goes, by law no one can be turned away from a hospital in the US when in need of emergency care. If you're in bad enough shape to be brought in code 3 on an ambulance, they're more concerned with keeping you alive. Insurance is billing's problem. It gets sorted out later.
One of my ambulance rides to the local hospital got billed to my nickname and not my full name, and the computers at the hospital didn't catch it. They billed me $15,000 for the ride. My wife about crapped herself. I called them and fixed it, and it wasn't an issue (PIF). I don't use this hospital for ANYTHING. They almost killed me in that one. I mean they were really bad. I spent 29 days at one stretch in that place.
The pharmacy tried to pull the same thing with one of the prescriptions I was on. It was $3,000/mo (not a typo) and the national chain pharmacy said it wasn't covered. I know I have good coverage, I I called them and had them check it again. After some typing and muffled discussion I was told it was fully covered with no co pay. I'm not taking that medication anymore. Anyway.
In all the times I was in an emergency situation as far as health care, billing came second. So while it would be nice to have my insurance card in my phone, it's not a must have.