I'll need to post this in Repair..
This is an issue with the Shimano internal hub, rather than the bike or any assembly work done on it, I think.
Adjusted the shifter to try to tune it, as advised. I ended up bringing the shifter out to the maximum extent available.
Now I have three gears: 1 (Usually first, occasionally second), 2 (sometimes second, sometimes third, about half and half), and 3 (always third, still).
As far as the gears are concerned, I got the default (44) chainwheel rather than the smaller option.
Unloaded, second is a good cruising gear. Third is on the brutal side. Loaded, it's easier to stay in first a lot, so that I can spin and not mash, even if the gearing isn't perfect for that.
But i'm in New Orleans, which is table-flat. First gear isn't what I would consider a good climbing gear with a load. It's not horrible for accelerating through intersections, but if you have any hills, spring for the smaller chainwheel set.