The engineering team refused to take the less expensive route, and insisted that the mute button physically disconnect the circuit, so that no future engineering team could decide to stealth "unmute" the microphone through software.
To this day, you can disassemble an Amazon Echo device and you will find a physical disconnect of the mic circuitry when you push the mute button. Don't want an "always listening" smart speaker? Just keep it muted, and a red LED circle informs you that the mic is physically disconnected.
I'm proud of the approach that Amazon takes to privacy. Privacy of customer data is considered the most important thing to Amazon, and this customer obsession (the #1 leadership principle) permeates the organization.
Disclaimer: I'm a principal engineer at Amazon.