From James Hoffmann he recommends 2 minutes. If I recall correctly waiting longer did slightly improve the results but after 2 minutes you get a good brew. https://aeroprecipe.com/recipes/james-hoffmann-aeropress-rec...
I like the button idea. 1. I'd love to see it battery powered. 2. I found a vape thing yesterday - I would love to see a battery from a Vape re-used if possible. 3. Lastly - based on a project I'm trying to do now. Can you somehow sleep the microcontroller so it uses no power while in standby and then starts the timer, so you have close to indefinite battery life?
Regarding the last point, you can do better than sleep (lower power state). You can have the microcontroller cut its own power once it's done its work:
https://randomnerdtutorials.com/latching-power-switch-circui...