As soon as I posted the last log, I realized what the problem was. The TPICLK wire was loose all this time. After fixing that, programming with AVRdude and USBASP went without any problems, and I could test my music-playing code.

Of course it didn't work properly at first. I had to change the timer initialization a little, use a different prescaler and adjust the octave. I also decided to run the timer at a higher frequency, because it's a 16-bit timer, so I can get better accuracy this way -- so I'm actually shifting the counts for the tones right a few bits.

But I still got some very weird glitches here and there, and I couldn't quite see the problem, until I realized that they happen randomly, but only when a higher frequency sound goes after a lower frequency. Finally I realized, that when I'm changing the timer's match register, I'm not resetting the counter -- so if the counter is already past the point to which I'm setting the register, it will go all the way up to the full 16 bits before it resets. That also happened with the 8-bit counters on the other chips, but it wasn't so visible, because the distortion was very small. With a larger timer it became much easier to hear. A fix is trivial -- just reset the counter.

And that extra code for resetting the counter took me exactly to 500 bytes of compiled code:

AVR Memory Usage
----------------
Device: attiny10

Program:     500 bytes (48.8% Full)
(.text + .data + .bootloader)

Data:          0 bytes (0.0% Full)
(.data + .bss + .noinit)

I guess I could put more songs in there...

In any case, now I'm just waiting for the PCBs to arrive. In retrospect, I could have made those new PCBs even smaller than currently -- with the puny ATtiny10 and a single resistor they are almost empty. But the piezo speaker takes some room, so I guess this size is fine.