There is an interlock pin in the film back, at the lower right of the film insert. There is a corresponding pin on the body (which sticks back from the back surface of the body by a very small fraction of a millimeter.
The insert/back's pin needs to protrude and push against the body's pin , in order for the body to sense that the film is advanced properly. If the body does not sense film-advanced status, it will continue to wind continuously (unless the multiexposure lever of the ETRSi is activated), the same behavior as if no film magazine was mounted on the body.
When film is not advanced, the pin in the back is recessed within the chrome sleeve which protrudes from the film back. When the film is advanced, the pin in is about flush with the chrome sleeve (or sticking out a small fraction of a millimeter) to push against the body's pin.
It is possible that either you have two bad inserts which do not protrude their interlock pins far enough to sufficiently push the pin in the back which activates the body-- when the film has gotten to frame 7 or 8, but it is fine before that. Or it is possible that your body needs adjustment. Or both.
If Bronica were still in business and their repair department was operating, they would tell you to send both the offending backs as well as send in the body, so that they could check and adjust as appropriate.
Since you state that the problem appears with a second body, that is a pretty good indication that the film inserts/back are in need of servicing, not the body.
I once had a problem with either of my new ETRSi bodies when used with a used back which I had later purchased. This caused the camera to not fire the shutter intermittantly, but only with the one used back, not with any of my new backs or other used backs which I owned! (I had multiple backs for professional shooting at events, etc.) So Bronica told me to send one body and offending used back, so they could adjust and verify proper operation. In my case, the pin in the back protruded out too far into the body sometimes, causing the pin in the body to sometimes cause the reflex mirror to not swing upward the entire way, so the shutter would not open. They shortened the pin in the back, and all was fixed.