That wall switch is not a simple on/off contact switch. As you can see, there are multiple functions on it (light, door control) which are represented to the opener via some sort of encoding. A simple surface mount doorbell button will work instead.
I avoided involving the hardware door switch (I also have a Genie opener) by buying a cheap Genie compatible remote and hooking that to the relay. Here is the one I got:
https://www.ebay.com/itm/123963473925
I also use the Zooz Garage door app to combine the remote and the door open/close sensor into a Garage Door Device
You will also need an open/close sensor (either a hardwired contact switch through the ZEN17 or a Z-wave or Zigbee contact switch) to determine if the door is closed or not.
