Thanks for the detailed writeup. The first thing that comes to mind is that your opener may be trying to run excessive current (1A or more) through Garadget’s relay which triggers it’s protection circuit (PTC Thermistor Current Protector Resettable Fuse). Excessive current is not unusual in very old openers because they used to power high draw circuits in wall consoles.
If you have a meter, you can set it to DC amps and short the terminals through it to see how much current is running through the button.
Since it works sometimes, the current must be close to the maximum current that fuse can handle. If that’s the case, you can bypass the PTC (surface mounted component located next to the relay on the PCB) by shorting it out.
If the current is too high it can potentially fuse the relay’s contacts. Before upgraded relay and protection were added, this happened with some real old openers and the workaround required a beefed up external relay.