From your description, Pin 8 on J1 is a “common” pin for the joystick switch and the rotational signals. Probably the ground reference, but possibly a power supply voltage (whatever voltage is used). The wide trace reinforces that theory, I believe.
Based on what you’ve determined and the note about the waveform on pins 6 and 7 of J1 to pin 9, I would guess that pin 9 is either a power supply or possibly a ground reference for the device inside the joystick which generates the wave forms. I’m pretty certain that to imitate the rotational signals, you’ll need to be able to produce two synchronized out of phase sine waves on pins 6 and 7 of J1 of the proper amplitude relative to pin 9. The phase difference would allow the receiver to determine the direction of rotation. That’s not something easily tackled, even by an experienced electronics design person.
To confirm that this theory is correct, you would need to use an oscilloscope to view both signals simultaneously. Anything else would just be guessing, as I am doing here.