Nothing with WiFi will ever be coin cell battery powered. But that doesn't mean it couldn't be battery powered for a year or longer with bigger batteries. Otherwise you'll need a different radio technology.
Well, except for all the coin cell operated matter over WiFi smart buttons / remotes / switches you can buy from just about anywhere…
I’m not sure I’d want a full second of latency on button presses. I would have figured Amazon used BTLE to the Alexa and used it as a BT to WiFi gateway