Hi @blazefenix ,
As it is working when using the external power supply you would have to open up the console and check if the battery terminal wiring is OK between each of the terminals of each of the batteries to the next and between the two battery compartments and then onto the “power board”.
As there are 6 x 1.5V AA batteries they would need to be connected in series connection (example only - same principle for 6 batteries or more depending on what voltage is required as they add up) to get the 9V required for the console to operate. The positive end and the negative end of the “battery” i.e. 6 x 1.5V, would need to be connected to the board.
It would take only one broken wire or connection for the console not to work as the battery supply would be open circuit
Here are some images taken from the ifixit Sega Game Gear Screen Replacement guide to show what I mean.
The image on the left shows what I think may be the power board as the DC-In jack for the external power is there as highlighted. The image on the right shows the wiring between the battery compartments.
(click on image to enlarge for better viewing).
If the wiring seems OK and continuity tests with a DMM (digital multimeter) of the wiring proves that it is OK between the battery terminals and the power board then further testing would be required on the board.
Hopefully a start.