All interpreting assignments that I have done at doctors' offices, clinics, and hospitals, have been paid for by the doctors and hospitals. I've never heard of any of the Deaf consumers or their insurance paying for terps for medical appointments.
I've interpreted for Deaf patients, and Deaf family members of hearing patients. The doctors and hospitals paid for both. Their hospital rooms had captioned TV's, internet and email access, and TTY's.
Sometimes the hospital social worker has to intervene to get the services in place for new Deaf consumers who don't know how to advocate for themselves. But once they get set up, things work out fairly smoothly.
Of course, emergency situations don't always go smoothly due to the fact that they are EMERGENCIES. An emergency is something unplanned and unexpected, and is stressful for everyone concerned. It takes at least 30 minutes for the hospital to contact my agency, and then the agency to contact an available terp. If they contact me late at night, sitting at home in my PJ's, I need time to get dressed and drive to the hospital. That's another hour. So, from the time a Deaf patient shows up at the ER, until the terp shows up, is at least 1 and 1/2 hours.
Of course, some hospitals use video interpreters, so that might cut down on the wait.