I have game gear that switches off after one second, it has been recapped completely but nothing. I discovered something weird though. when I plug a tv tuner in it it stays on. I can see anything as there is no analogue signal anymore in my country but is stays on. So I can definitely say that there is no problem with the light or the screen as they both work. I hope that can help someone with a bit more knowledge then me.
I have the same problem but something weird happens, when I plug a tv tuner id stays on. It does not make sense ( to me at least), when I plug a cartridge it switches off after one second, the same without card, but with the tuner stays on and works fine.