Well, hack is a strong phrase, it was all physical with soldering, no software or firmware changes. And, if Amazon still listens while the mic is "muted", I'm no more private than before. Also, any phrases I say to it will still be treated the same.
My issue was that the mic button is annoying to use from across the room. The whole point of a smart speaker is it only needs talked to. No need or reason to walk over. So, having to get up and unmute to change the music is annoying. I'd rather not have a giant corporation constantly listening though. So, muting was preferable to not muting, but again, that defeats much of the utility of a smart speaker.
What if I could unmute from my desk, while the Echo is on a stand across the room?
I decided to take it apart because I wanted see how it worked, especially to see if the button physically disconnects the mics and if there could be inserted physical disconnect switches. After deciding that the button is likely disconnecting via software/firmware, and that disabling the mics would be more trouble than it's worth, I thought I might try to add another mic button that could be extended to anywhere in the room.
The top board in the Echo seems to contain most of the logic, and also has the buttons for actions like volume and mute. The issue is that the buttons are SMD. I do have a hot air reflow station, but I wanted to preserve the original button in place, not remove it and move elsewhere. So, I used a multimeter to discover one part of the button goes to ground, and the other connects to one of two tiny dots of solder on the surface of the board, right by the button. So, after many minutes of attempting to solder a dot smaller than a pencil tip and two strands of copper wire twisted together, using the tiniest tip I have, the connection finally stuck. Then, I led the uninsulated wire across the board, insulating with Kapton tape as I went. Switching to insulating wire, I directed it up the side of the chassis (which is later covered by the cloth and plastic outer shell). Nearly done, I reattached a ribbon cable from the main board to the power board, and plugged the Echo back in. It didn't work. Turns out, I had connected the ribbon cable incorrectly. Finally, it worked! Finished up by adding a sort of port to the outside of the echo for button use.