I would definitely power it from your bench PSU and set a lowish (reasonable) current limit on it say a 100mA. Apart from anything it'll tell you if anything if drawing power - just because it doesn't seem to power on, doesn't mean it isn't powered.
If it's not taking power try and trace the voltage around from where it comes in to where it stops. If there are IC's, see if you can look them up and find out which are the power pins. If not guess. Looks for any breaks etc or dodgy components.
I would think based on the fact that the batteries corroded you are looking for something physical, either corroded track or something. Try and poke around a bit and see if anything you do alters the current taken from the psu (no, don't just short the battery!).
If it is taking power when you first powered it on, but is in current limit. Power off and physically inspect for shorts. If yo can't find anything, steel yourself and crank up the current limit. At some point, if there is a fault something will start to get hot. Use the tried and tested "finger" probe. The hot component is likely your problem.
All a bit rough and ready, but should be a start!