To those of you with a DOA machine that makes noise and gets hot but produces no water:
I'm the kind of guy who takes things apart even when they're working, so the idea of sending this back to Breville untouched never even crossed my mind. I tried letting it run for a few minutes as recommended in another comment here, but that seemed to have no effect, although I'm fairly sure it's the same problem.
I took the bottom of the unit off, which requires taking off all the screws around the edge except those for the back "feet". I then removed the water feed tube from the water input. After verifying that, when placing the reservoir in the machine, water flowed freely from the little nozzle, that eliminated the reservoir and flow valve as a source of trouble.
When I turned on the machine (yes; water, power, bare wires, etc., I know), there was a clear feeling of negative pressure on my finger when placed on the end of the tube. I poured some water into the tube, placed the end in a glass of water and turned on the machine. It began to pull in water and eventually water came out of the steam/brew openings, after working the air bubbles through the system. Once the rubber tube was full of water, I carefully reattached it to the water source without letting it drain. Everything seems to work perfectly now.
My theory is that the machine is somehow not able, or only sometimes able, to clear the tube of air when it is first operated. Most likely, this is because the pump is designed for water rather than air. It may be that tipping the machine during operation could help this process along without the need to open it, and it may be that having the reservoir as full as possible will make it more likely to work due to the slightly increased pressure.
So, if you have a little mechanical inclination, and are the kind of person who makes insurance companies nervous, you can probably fix this problem in about 15 minutes.
Enjoy your coffee.