By showing numbers when you press buttons, the display board is showing that it is functioning correctly - it just is not communicating with the main board. And since the SQ-80 plays, that’s a good sign that the main board is working correctly, too.
The complication is that the communication path from the main board to the display actually goes through the keybed, so the culprit could be a bad ribbon cable that connects the main board to the keybed. And since you said the display shuts down completely after 5 minutes, there could be a power supply issue going on.
So for starters, make sure that fuses F3 and F4 on the power supply board are not blown. Test them with a continuity tester if you have one - you often can’t tell just by looking if the fuse is blown. And then make sure the ribbon cable to the keybed is seated firmly, and plugged in correctly. (Mis-plugging it will blow those two fuses.) Do all of this with the keyboard uplugged from AC power!