Nest thermostat causing condenser unit contactor to chatter when NOT calling for cooling
This is a hack, but....

Try replacing the relay with a much heavier duty one.  Specifically one with a coil that uses more power.  24v is a very common contactor voltage.

Here's one option from Grainger, but really any 24Vac coil voltage relay would work.  The contact rating must be equal or greater to what you currently have.  Aiming for very high voltage/current contact ratings, or relays with more than 2 poles will likely get you a higher-wattage coil, and that's what you want to reduce chatter.

The heavier duty coil will pass more current through without moving.  In fact you don't need to remove your current relay. maybe just connect another in parallel with it so its powered on at the same time.  Leave a note for why you did that, or the next hvac tech you have out might yank it and tell you how crazy the last guy must have been to put that there.

Or for that matter, you might be able to put any sort of load in parallel with it.  A suitably rated resistor for example.

I've used a nest for about 6 years myself in two different houses and what's happening to you is not normal.  It sounds to me like the 'support' you were talking to were grasping at straws.  I will agree with the other posters that Nest is junk, and you'd be better served returning it to the store.  Google will be discontinuing them soon, and you will lose control of your thermostat overnight against your will when they shut down the cloud component of it.

But since it's already installed, if you like it, there's how to make it not clatter.

There is also a small, but non-zero chance that the base plate of your nest is defective, or one of the connector pins on it is busted, and the c wire energy isn't making it to the nest itself.

Check the nest app or the thermostat's advanced/installer interface itself to see if it is AWARE of the c-wire's presence.  It should show you a photo of the wiring it detects.

If the nest thermostat itself (the piece that comes off the wall) does not detect the c-wire power, then it probably will "fall back" to stealing power from the other wires.

If both of your Nests have baseplates and you experience the same problem with both baseplates, then this is probably not the cause.  My first thought when I started reading your post was the c-wire was broken.  But you ruled that out with your voltage measurement.  Good on ya!