The 'smart' openers have a continuous voltage across the 'smart' button controller. The controller button has a clock and provides other functions. Creating a short across this circuit, with a parallel relay, reboots the 'smart' controller.

Last time I had a GDO replaced in my home, I opted for a 'dumb', old-fashioned model, to ease integration.

If you can remove the fancy pushbutton controller (which displays the time) and replace it with a simple pushbutton, I think your relays should work. Not sure if that is an option for you.