That area is for the MCP, JTAG and a few CPU signals. A lot of the pins are no connection or ground. I have seen non-liquid damaged boards with this area looking like yours, and they have been fine, so I don't think that is the cause of your issue (just age).
You can use flux and solder the connections, then wick it if you think it will help, but I don't think this is the cause. Get a schematic and board view, use these to measure some voltages.
Where was the original liquid damage? Check in this area, could be some shorted components or burnt pins. Feel the chips on the board whilst it is plugged into the charger, to see if any are getting hot.