By George, I think I've got it! (couldn't resist). Remember I'm using an adafruit test program to test an adafruit clone 16 channel servo controller. The test program has a comment that says "called this way it uses the default address or 0x40." Hmmm. In 40+ years of programming, I've been caught by accepting defaults before. So I put in the 0x40. Voila! Everything works.
I'm just suspicious enough to wonder if the default only works on "real" Adafruit boards, but I can't see how they'd know.
Would all the people who told me I wired it wrong please stand up :-).