This was either a carburetor or a fuel supply issue. The carb is factory sealed, so I was not going to void the warranty to fix it myself. I had the mower picked up for a warranty fix.
I just got a call today that it wasn't a carb issue, but a fuel supply issue. There was a bunch of crud in the fuel lines, and the filter was 100% clogged. I hadn't thought to check that, because the thing had hardly been run, and why the heck would there be gunk in the fuel tank, even if it had been sitting for 2 years? Anyway, after getting a great deal on the mower ($900 for a $1600 mower), putting $75-100 into getting it running is a small price to pay!
Overall, still a good investment!