Without giving any particular advice, I'd say your initial hunch about battery leak damage is most likely correct.
Check the ICs right next to where the battery was. IIRC there are a lot of address demux ICs there, and it those traces are damaged, or the chip it self, all kinds of weirdness happens, which is probably why the battery leaks tend to lead to all kinds or bizarre problems. Like IC 30 and 31.
Check the traces for continuity.
Check vias for continuity.
Traces also run under the ICs so there could be damage you won't see until you remove the chip. Damaged vias are obviously difficult to see as well.