If you have access to a multi-meter you will have a better chance of finding the fault as you can measure actual verses expected voltages at various points in the circuit.
Without that there is not much you can try, if I understand the issue correctly it is that the Lynx turns on as soon as a power source is attached and does not turn of until it is removed. Of the top of my head there are a few of primary things I would look at...
1) A short circuit of flex circuit/stuck power On button
2) A short (or bypassed) Q12
3) A problem with the On/Off power latch comprised of U6, R70, C38 and the Power On/Off buttons
Things you can try without access to a Multimeter...
1) If you press the Off button when a game is running does the game restart as though the reset button had been pressed? If so that my suggest that the lynx is powering off, just powering back on again too quickly for you to see it, possibly due to a stuck/shorted power On button .
2) Short out C38 (or connect pin 11 of IC 6 to pin 7 of IC 6) with a piece of wire. Does the Lynx turn off with the wire in place or remain on? If it does turn off off does it immediately turn on again when the wire is removed?