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.
I think some do use regular wifi, especially the amazon ones, but with a button you have an advantage where you could just stay idle and off the network most of the time, and only connect to send an event when pressed.
That likely increases your latency by a fair bit. Most of my devices take a second-ish to connect, configure the WiFi stack and start sending data.
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
BTLE and/or their IoT mesh network (Sidewalk?) would be a good way to go too, but if it's just a button to reorder laundry detergent or something then I don't think even a 30 second delay would be a deal killer for most people.
I was happy to make it work at all :-) There is a guy on the internet who measures the power consumption of many mcus and concludes the nrf52840 is the best: https://www.youtube.com/watch?v=KE7bOYCYETM
But ultra low is not necessarily the same as battery powered, which itself doesn't necessarily mean coin-cell battery powered.
My experience with wifi modules I've built so far has been, they've all been "low" but not usually battery powerable.
I wish the README provides more info whether it's suitable for battery-powered operation, and if so how which.