My colleague Greg figured out why my 74hc163 counter IC wasn't working. The problem was not the counter - it was the 74hc04 IC I was using as an ersatz logic probe. With the 150 Ohm resistors I was using, the LEDs were consuming more current than the 74hc04 could supply.
Greg gave me a ULN2803A peripheral driver IC which is capable of sinking 500 mA per pin, which is more than ample for lighting an LED. I modified my "logic probe" circuit, and now everything is working nicely. As a side-benefit, I now have 8 inputs rather than just 6.
Now the 74hc163 properly rolls over from 1111 (15) to 0 (0000). Next experiment - cascading two 74hc163's to make an 8-bit counter.