I love butterflied & fried quail breast, but I like the legs just as much, Y'all don't keep the legs?
We clean the the same way pretty much. Cut off head, wings & feet at the feather line, peel off skin, now lay the bird in palm of hand, breast down, with it's rear end facing you, now cut the legs off where they attach to the back bone with shears, flip bird over & cut the breast out with shears all guts stay attached to the carcass.
I always save the legs. Ask Big Joe about the quail I dressed for him. I hunted quail for around 30 years before I found the method I now use to field dress them. I'll show you rather than tell you my method the next time I have a camera available. Here's the best I can tell it with my limited writing skills:
I clip off the wings and the legs at the feather line just as you do, but I leave the head attached. I skin the bird, and hold it in my left hand as you do. I then grab the left leg firmly and pull it over the body, and back with one fluid move. It pulls right off. Then do the same to the right leg. That leaves me holding the breast with the head and the backbone. I grab it by the head and neck, then pull it off, which leaves me a breast in my left hand, and the backbone with guts attached in my right hand. I can do this almost as fast as a helper can cut off the wings and legs.
I generally bring them home this way, then butterfly the breast. I'll take the femure or thigh bone off the leg.