Desoldering the three chips gave me access to the signals that control the LED matrix:

1. Row control, bit 0
2. Row control, bit 1
3. Row control, bit 2
4. Row control, bit 3 for green LEDs
5. Row control, bit 4 for red LEDs
6. Column control, serial clock
7. Column control, strobe
8. Column control, serial data

The numbers on the list above match the "resistor color codes" of the wires I soldered on the board:

At the other end of the wires, an Arduino Micro:

My objective for now is to be able to draw something with an Arduino at least. Since it's all 5V powered, that seems the easiest goal for now. I plan to reach that goal in three steps:

1. with a row powered directly by power supply, verify that the shift registers are used correctly;
2. with a single column turned on, verify that all the rows can be activated
3. try it all with some bitmaps or text

I've managed to do step 1, after finding a small mistake in the wiring (as the purple and gray wires are really similar in color in these wires). I can now control each of the LEDs of an individual row:

The thing draws 5A, that's 45W of power, same as specified in the power rating behind the case. That bridge rectifier is too hot to touch, but I think it can stand that.