I like your idea of cleaning the relay's contacts if it turns out to be bad, then throwing the cleaned relay in your glove box as a spare. You don't want to trust that relay (metal pieces everywhere, yuk), but as a spare? Absolutely. As a prepper, two is one and one is none. Good advice.
You should see the spares I carry.
I'm glad you brought up the diode, it completely slipped my mind. If you've got those probes in the correct direction, the ohmmeter should have enough voltage to cross the PN junction and it'll show continuity across the diode even if the coils open.