What is this filter switch doing to the AC line that triggers the switching power supply to function and power the rest of the unit?
What is this filter switch doing to the AC line that triggers the
switching power supply to function and power the rest of the unit?

It's probably switching the neutral line hence it is still disconnecting power to the device but (unfortunately) it's still leaving the live wire connected. I guess you measured the presence of live voltage with respect to the chassis or earth connection.
BTW, it's not a filter switch; it's a switch.

Considering the AC line is always live, is there a benefit to this
method? Versus, just using the switch to completely cut the power.

It's switching the neutral connection and this is not ideal but it will disconnect power to the circuit.