Yes, apostrophe's led you a bit astray. The stuff in your photos is just hot glue, used to secure the components in place.

The residue from electrolytic capacitors is not that voluminous. It's mainly visible as a dirty smudge on the circuit board, immediately surrounding the capacitor, but not generally obvious on the capacitor itself (except, perhaps, on its underside).

Your problem is almost certainly from leaking capacitors on the logic board. Yes, by all means check ram and rom (because it's easy to do so), but you will almost certainly find those to be fine.

The only true fix is to clean the board, and then replace the electrolytics in question. It's not hard (just time-consuming). And if you use good caps (tantalums), the repair will last a good long time.