I can confirm that the HT16k33 chips I had were most likely faulty. I took one of the remaining unpopulated boards, and carefully checked it for shorts and continuity. It checked fine -- each pad is connected with exactly one hole. Then I took my last remaining chip, and soldered that carefully to the board. Added the pin headers, and plugged that into a breadboard. Then, without connecting any matrix, I carefully connected the wires for power and for I²C (with extra pullup resistors on them), and plugged the whole thing into USB.

Then I did the I²C scan, and the result is the same as with the other chips. I double-checked the datasheet if there is any need for additional components of any kind for this chip (and also examined the breakout board I have for any extra components). Nothing. It should just be working, and it's not. Ergo -- broken chips.

I ordered a new batch from a different seller this time, and will see how that works. In a few weeks. Perhaps the new PCBs will arrive by that time too.