I have no idea why the piezo doesn't cut out when you operate the switch, it certainly should do. All I can think of is that the switch may be faulty or you may have a strand of wire bridging across the contacts so it's always made. You really need to check the switch with a multimeter. And if wires have broken several times when you've put the harness in, then it could well be possible that a strand has pulled out and shorted across the switch contacts.

These days I always use heat-shrink on connections wherever I can, as it both prevents accidental shorts and provides some mechanical strength to the solder join.

For the piezo-only operation, it looks like you'll have to do the alternative volume pot connection method and swap the blue and red wire connections over on the two volume pots. That means that the taper will be modified a bit, but you'll always have 500k resistance between ground and the signal connection for a single pickup selection, and 250k between signal and ground if both pickups are selected. Both will be sufficient resistance to keep the piezo signal happy.