Update.
I tried to replace all transistors, electrolytic and ceramic capacitors. I checked the diodes (which are functional). Now you can hear the sound (very very low so I'm not sure if it didn't work before), but the screen won't turn on. In the meantime I have designed and 3D printed the joysticks, which are perfect.
I keep looking for a wiring diagram. The only thing left for me to try is the screen (Futaba DM-20 VFD) which doesn't look damaged, but I don't know how to power it directly.