I thought that emergency responders don't have the time to check people's phones, and as a matter of policy don't. And even if they did, there is no guarantee that the information there pertains to the injured person, who might be carrying someone else's device - with potentially disastrous implications. So, and I could be wrong, my understanding is that they completely ignore the phones at the scene of response.

I think it's a good idea to have this specific medical information handy, but not for that particular purpose.