Any additional grounds will help, but first are you sure that the cutout in the dash is clean metal?
Check the wire where it connects to the pushbutton by using a jumper there at the connection. Check it to several different grounds including the dash and see if any of the grounding locations work. If it works on one of the ground locations, it is the pushbutton itself or the ground from the pushbutton to the dash. Hard to believe the whole dash is not grounded, but worth checking.
If none of the ground locations work there is a problem in the wire itself. As already pointed out, an ohm's meter would help check the wire end to end, the pushbutton itself and the pushbutton to ground.
If you don't have a voltmeter to check ohms, you can get a test light with a battery in it and a light that will show a good electrical path.