Lots of other good ideas here. I’ll just add some thoughts.

Adding some pipe insulation from the hot water heater to your shower might help a bit.

I suppose you could add some instant hot water heater system or even a second small hot water heater close to your shower to eliminate the heat up time.

If you are this interested it might be worth having a plumber come out to consult about what the issue might be and what the possibilities are. A lot might depend on your specific setup and how accessible the pipes are without completely ripping apart your walls.

How does it open? Like a door bell button? I think MyQ has addon things that will use that to open/close it. Wyze might be the cheapest you attach their garage door device to a Wyze camera and point it at the inside of the garage door. It pushes the button to open/close and uses the camera to sense a QR code you attach to the back of the garage door to tell whether the door is open or closed.