The traps you describe would tame most of the flutter echo and room modes. You can try this for yourself in another bedroom that isn't treated. Just stand in the center and clap your hands. Putting a mic in that position doesn't "stop it at the source", if anything it amplifies the problem. Ever seen a parabola mic? Like the ones they use at sporting events? They can pick up sources across the field by concentrating the soundwave at the mic. Micing in the center of a room that is nearly a cube does the same thing. It focuses all the flutter echo and other audio artifacts right into the capsule.
The best place to set up a mic is in a properly treated room, the bigger the better. As the op is finding out, condensor mics are extremely sensitive and pick up whatever is going on in the room.
I tried the moving blanket thing years ago, and wound up building gobos because the blanket thing just doesn't work. The solution is to properly treat the room, then pick one end or the other-- preferably with treatment behind the mic too. Or find a better room. One that isn't square and has a vaulted ceiling, and treat that room appropriately.