Since the board schematic is presumably proprietary, and the board could be multi-layered, you'd have to do some fairly destructive investigating in order to confirm that the switch is actually doing what it says it is doing.
Not as long as it cuts the wire that breaks all connections between board and the microphone(s) (and board is somehow audited to not contain any extra microphones or otherwise sound-sensitive components).
<tinfoil hat on>But this is only as long as you actually can verify or trust that the switch is actually a switch and not a device that pretends to be one. A transparent casing where one can visibly confirm the actuator operation is a good idea.</tinfoil hat off>
If you want extra tinfoil hattiness, most multilayer ceramic capacitors (the surface mount kind that are everywhere on PCBs) are somewhat microphonic. Condenser microphones, especially electrets are basically just capacitors designed in a special way to maximize this effect.
> If Amazon advertised it as a physical switch, and it was not, that is fraud and actionable.
A physical switch is still a technically a physical switch, even if it's not connected to anything. It would come down to how they are describing what it does, and what guarantees they make (or not.) Do they have any, or is this discussion entirely hypothetical?
They need to advertise it as a physical disconnection of the microphone.
The switch doesn't need to be part of the main board --- it could be located next to the mic, so it's trivial for anyone with even the most basic electronics knowledge to see that it does what it says it does.