My insurance at work has put out a notice that this is the problem and the misleading statement in stand-alone ERs. They do accept your insurance but the stand-alone is usually not in the insurance network. There is no financial incentive for them to join the network since people will worry about the insurance later if it is an emergency or even urgent. The hospitals tend to join insurance networks because of the other care they offer and people having more choice in it (even in the hospital stay, but a lot of the tests they offer can be done outside the hospital too.) I understand that you need to check the doctors separately, along with the labs and other independent contractors tot eh ER (anesthesiologists, x-rays, etc.)
I have not used a stand-alone ER yet. My local hospital is so rural that there is almost never a real wait anyway, and they can stabilize you before transfer to a better hospital if you need it. If something happened to me where I needed an ER at work, I am about three blocks from the big county hospital where there is a wait for the ER but the worse your condition, the higher priority you are for service.
I would probably consider a free-standing ER if there were one in my area and if I had checked prior to use to make sure that both the doctors and unit itself were in network for coverage.