Now that I know the ATtiny10s work, it's time to publish the project! I am still waiting for some common-cathode RGB LEDs to arrive; until then I can just test one channel at a time.
The most recent KiCad PCB is actually a little different than the physical PCBs in the photos. I originally routed power to sides of the LED, but that's easy to fix with a little hidden rework on the underside of the PCB. The lovely Oshpark After Dark front side is unharmed!