Decades ago, I had a brand new body and back, and I purchased a used back. The behavior (with the used back) was that I would cock the camera, advancing the film. When I pressed the shutter release, the mirror would flip up, and SOMETIMES I would see the red light blink but other times it would not blink. And when the red light failed to blink, the frame would NOT BE EXPOSED. Bronica told me to send them both body and back (and no lens). The 'film advanced' pin was TOO LONG, so it would somehow not allow the reflex mirror to flip up fully, and the shutter would therefore not fire! They shortened that pin, and everything worked properly.
If you have another back, try out the body (and lens) with that second back.