This is the first of two significant observations.

First, since the GPs draw a lot of current, the overhead cabin light should dim. When the the GP relay did not pull in to power the GPs, the lights did not dim.

This observation is more significant and raises suspicion of an intermittent 'bad' connection in the power wiring to the relay itself, tougher to track down than a complete failure, but not impossible. Let's suspect a loose, intermittent connection somewhere between the 'ignition' switch and the relay.

Having a helper for this part is, well, helpful! While you may be able to hear the relay under the hood 'click' while sitting inside the cabin when you turn the key, it is easier if you are able to stand near the front with the hood open. You can hear better and you may be able to feel the relay click by placing a finger on it. Cycle the key on and off a several times, listen for the click and observe the cabin lights each time. No click and no dim lights in the cabin = no relay action. Sometimes click and dim means a sometimes bad connection.

Turning the key to ON applies voltage to the GP relay, and this voltage stays on as long as the key is ON. You can test this by using a test lamp or a voltmeter to probe the correct pin socket after you unplug it from the relay. When you turn the key you should see the test lamp light, or read 12V on the voltmeter. Again, having a helper to turn the key on and off in the car is easier, but you can do this solo if you place the test light or meter where you can see it from the driver's seat.

If you never get no test light or meter indication, the relay circuit is broken. If you 'sometimes get an indication, there may be a loose connection in the wiring or a bad switch contact. Try 'jiggling' the key when the light or meter is not reading, and if the light then comes on you can suspect the contact in the 'ignition switch' is going bad.