The shutter may not be capped when when it cocks, but the mirror would prevent most light from getting through - which might be why a shop didn't catch the issue. This would explain the "sweeping" effect of the light leak. The MEs are apparently known for shutter issues, but likely it's just sticky from age.
One experiment would be to put a lens cap on when advancing the film. If the light leak between frames disappears then this is certainly the cause of it.