The physical mute switch -- that is, the piece that you move back and forth with your finger -- is not part of the cable assembly. The flex cable has a tiny internal switch that sits in a recess under the physical mute switch.
Although it's not in English, [link|https://www.youtube.com/watch?v=FQ6Vs-Y589w|watch this video|new_window=true] starting at 3:04 and you can see how the assembly comes apart, and at 4:08 in the same video you can see the technician separating the flex cable from the physical mute switch, giving you a pretty clear view of how they mate.