Genie eh - did you try “open sesame”?
is the existing wall switch/controller, 3 buttons? It won’t work in a standard way with GD00Z-04. I’d call that switch a controller, since it is not a simple switch (on/off) - it actually uses at least 3 states to signal the opener (open, close and light). A bit too complicated.
The Linear GD00Z-04 is made for interface to “dry contact” input on garage door opener - it’s an old standard of “push the single button to open”, “push the single button to stop”, and “push the single button to close”. THAT is pretty simple, in this domain.
If you take the controller or a wireless remote apart then you may be able to solder additional connections to achieve what you need, but it’s not trivial. I’d try jumpering the main button connections in the controller, before soldering, to see if you can make it send the signal you need.
It’s a crap shoot. Good luck.