I don't know if in this case the shielded wire should be connected at both ends. From what I recall the shielded wire should only be connected at one end or you end up with a ground loop.
The footswitch don't work like that. It doesn't carry a signal, per se.
Instead, the trem switch is connecting a wire to ground (originally to enable the oscillator to work). On the reverb side, the switch kills the reverb signal by grounding the grid of the reverb recovery amp. But since that wire is connected to a live tube grid when the switch is open (not grounding), the wire needs to be shielded to avoid hum pickup.
It's fine to attach the shield wire to the case of the pedal and the plug, because these should be viewed not as "ground" but as continuation of the metal shield.
It just so happens the shield is also grounded at the chassis end. The metal switch housing doesn't count, because it is only "grounded" by virute of the connection through the cord shield back to the chassis and "real ground".
But doesn't the footswitch sit on "ground"?
Yes, but is has a plastic bottom, and "ground" of the floor might be a slightly different potential than wherever the ground spike is that is ultimately conencted to the ground buss at the service panel, and on to the amp's 3rd power cord wire.