A quick PCB surgery, and I cut the trace from near the pin 1 of the pi connector from display's vcc, and connected that to the nearby 5V connector that I left there for power. That solved the weird behavior of the display and the restarts, but it's still making chirping noises. The images below show the changes:

In other news, I managed to move the whole configuration of the display into an overlay, and also got the keys work with gpio-keys driver -- except one of the fire buttons, which uses gpio 7, and which is a second CS pin for the SPI controller. The SPI driver just won't give that pin up.

I already started designing second version of the PCB that would use other pins -- tested working -- but I'm still not entirely sure if I'm going to be ordering that. De-soldering this display now would be a little tricky without hot air. I guess it all depends on whether there are more mistakes to be found in the audio system, which I didn't test yet.

I'm also still thinking about bet option for powering this. Right now I'm leaning towards a flat lipo sandwitched between the two boards, because that would be small, but it's really getting hot in there, so I'm not sure.