I like them on else I get visual sensory over load with all that stuff. I like the idea of the flannel on the door so it can be used as a display wall. I am also a bit of a freak about dust so the doors would hopefully cut down on some of that. Very nice closet area by the way!
I am also not sure if anyone else has this but I do not always like everyone to see and commend on my stash when they walk into my sewing room. I do have people in there all the time and I have a Quilt Bee at my house weekly.
It is probably all a personal prefference thing. It seems like you are really enjoying it open so maybe go for that right now and consider closing the windows off to shut out sunlight when you are not in the room. That is is if you are concerned about that. the doors would definitely make it feel less accessable since only half opens up at a time.