If the battery leakage ate into any board traces or solder pads then they would need to be jumpered for a reliable connection. If there is still residual corrosion or contamination on the board it can provide a path for extraneous electrical current to flow and that might interfere with some of the circuits. If the board was mounted vertically sometimes the leakage can go further than the immediate area when it drips, runs or falls downward.
Since the corrosion was so little maybe it is just one thing that is causing the problem.
The vinegar will neutralize the battery leakage and help prevent further damage but it may not remove all the dried contamination that's already there. You can try a more vigorous cleaning using a fairly stiff bristle brush and dish detergent, then rinse & dry well. Some people even put them thru the dishwasher before giving up. If the battery leakage got underneath chips sometimes it is unseen, we've removed chips to find nasty stuff underneath. And sockets and connectors on the board can get affected if the leakage went into them. There are places (some are people here on NLG) that will clean and repair boards, but you'd need to balance that cost with just getting another board. We've cleaned boards in an ultrasonic tank filled with water and liquid detergent, similar to a jewelry cleaner but larger. The vibrations do a good job shaking loose contamination and cleaning it. But the vibrations can damage mechanical parts on the board like crystals so it is a risk. Engineers recommend suspending the board in the liquid so the board doesn't touch the sides or bottom of the tank, minimizes the vibrations conducted directly to the board.
As cowboygames said it gets down to how much time you can spend working on it, and whether you just want to be done and get a replacement board.
our shop had something like this, we had a frame that suspended the board in the cleaning solution vertically or horiz so it didn't touch any part of the tank: