It probably depends on your area. Where I live, we used to have a VA Hospital, but when it was here, and someone requested transport (from the ambulance) there, we had to call and talk to a doc at the VA. If they denied it (99.9% of the time), we transported to the regular hospital (and the VA paid for it). If we didn't call or they didn't request, patient had to pay. Usually, the people that went to the hospital themselves, unless it was minor, were sent to the regular hospital via ambulance transport, or to the bigger VA in Pittsburgh for treatment. And there was nothing fast about it. We would often show up at 7PM and find the person had arrived at the VA Hospital around noon.
Without too many specifics, most major physical or mental procedures went to Pittsburgh. And minor major stuff went to the local hospital. However, now the VA Hospital isn't a hospital anymore, it's an urgent care center (no ER). I'm sure there are better run systems and worse runs systems (Cleveland Ohio's VA is huge and reminds me of an airport terminal). Really depends on the area and how you want to use it I guess.