The AD board controls a few things, from lower voltage , to signalling the backlight to turn in, to actually displaying the colors and imagine in the screen .
The smaller board with the blown capacitors you pictured is the backlight inverter board, which essentially gives power to the lcd portion of the screen. So if that's blown, it won't look like it's "turned on". It could be more is blown in the AD board, since that's what gives the inverter board both the signal to turn on and the voltage, but hopefully you replace those capacitors and that's all that blew .
The game will play blind without the screen working. In fact, if your machine has a normal lcd cable connection, you can just plug in any lcd go see it still working. Hopefully it works out for you!