@Jonathan Beri asked in the comments why I didn't use PCA9685, but instead went with an ATmega and my own code. To be honest, this is mostly because this project is the continuation of my #Servo Controller project, in which I used a Pro Mini board.

But that got me thinking, and I went and checked if I can get some of those chips cheaply, and I went ahead and designed a PCB for them:

I was actually so happy with it, that I went ahead and ordered them from OSHPark. Today the boards arrived, and I assembled one and tested it:

It works perfectly fine. You only get 16 channels (that's why I could fit it on the standard D1 Mini shield), but you can stack 64 of those boards (the address selection jumpers are on the bottom). Pretty neat.