Just in time for the contest, the boards from OSHPark arrived today!
I immediately assembled one in order to test it.
During that process, I learned about a new revolutionary discovery by the Chinese scientists! In their secret laboratories they developed a new kind of copper wire, one that is utterly impossible to solder, no matter how much flux you use and how hot the soldering iron is. Apparently, I got some wires like that from them, possibly by mistake. Using wires from a different vendor made it finally work (I also had to remove the speaker from the clock pin during programming).
Once I finally managed to flash the code, I only got clicks. A quick inspection revealed, that I used wrong pins for the speaker and the LEDs! The code expected speaker to be on PB0 and the LEDs on PB1. But the PCB puts the speaker on PB1 and the LEDs on PB2. Changing the LEDs pin is trivial, but can I change the speaker pin? Turns out that yes, the PB1 pin also is connected to the timer, and I vaguely remember changing the pins while designing the PCB with this in mind. I must have completely forgotten about it afterwards. In any case, just changing COM0A0 to COM0B0 fixed the problem. I also fixed a small bug where I only cleared TCNT0L and not the whole TCNT0, and it works.
The last bugfix made the code 2 bytes bigger, but it still fits within the limits:
AVR Memory Usage ---------------- Device: attiny10 Program: 502 bytes (49.0% Full) (.text + .data + .bootloader) Data: 0 bytes (0.0% Full) (.data + .bss + .noinit)