Are you in a rural area? I've always had a choice, and we have had some really crappy insurance plans through the years. I pretty much have to drive into suburbia for medical care, but there are three hospitals at about equal distance from me. All three accept my insurance and I have even more choices if I extend the radius out a bit more.
There's a real battle for patients, especially patients with good insurance, in this area. Smaller and rural hospitals tend to lose out and end up closing because they can't offer on-site MRI and other diagnostics and facilities that they don't really need for their patient volume. My part of the county just lost our only local (within 15 minutes) hospital because of that - people stopped going there for non-emergent care because being transferred to the next-closest facility in their system for advanced testing or specialty care was so inconvenient, and the ER was a money loser. So now the building is a walk-in clinic and birthing center, and the closest ER is in the nearest full-service hospital, about 45min away.