The cable release linkage on the Mamiya Press/Super 23/Universal is a rotating piece that sits behind the lens/shutter, attached to the lens focusing mount, not the shutter. The cable release pushes on a lever on the photographer's left of the lens, which causes the whole piece to rotate so that a pin on the photographer's right pushes against the shutter's trip lever and fires it. You should be able to observe this pin moving when you flick the cable release.
Problems triggering the shutter are sometimes because the lens+shutter has rotated slightly with respect to the focusing mount, so the shutter trip lever isn't being pushed far enough.
A different but related problem is that sometimes the manual shutter firing lever (allows you to fire the lens without the cable release) rests against the shutter trip lever and prevents it from coming all the way back to the unfired position. This will prevent the shutter cocking lever from catching. So if you ever have a lens that can't be fully cocked, it's usually because something is pressing slightly on the shutter trip lever. Remove that pressure and it should work again.