I'm gonna guess that the relay you're hearing is the small one towards the rear. That relay serves to turn on the coil in the big relay, near the center opposite the two coax connectors.
Have a look through the plastic enclosure of the big relay and see if it also goes "click" and the mechanism moves in a way that's clearly visible.
If this relay is not being activated, the symptom would be just what you're seeing.
Bad news is that this is typically the fault of the small relay, one contact point going bad.
But if you're really lucky, the wire that leads from the small relay to activate the big one has just come loose.
You hope.
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