Low power servo button pusher

Thank you for this, it's super helpful.

Since this is the first time I'm trying this, I think I'll get the clunky relay module (and then upgrade once it works as expected).

Which is the cheapest board that could do this? I guess Pico is great cos it's tiny.

I ideally want to run a timer on the board and then at certain times, activate the servo.

To help save battery life, am I right in thinking I could switch the power pin of the servo (without blowing it)? Do you have any articles that explains this in more detail?

I've seen this which seem to show a super simple approach:
https://www.wilsonleite.net/espruino-servo/

The adafruit site says "Never charge or use unattended.".

Is there a workaround for this, maybe safer batteries?