I went to a talk by a paramedic fairly recently. Presentation on cycling-specific first aid. Pretty good really and a worthy back up to the industry-standard First Aid at Work thing.
Bottom line is that paramedics don’t care about that, they’ll work on saving the persons life first, then worry about who they are later. And even that is more a job for the police.
Most smartphones, you can set the home screen to have ID or an emergency number, thus negating the need to get through the ever smarter security codes. ICE was fine back in the days when to lock a phone you pressed * and # together, much less useful now.