The leaky caps can cause extra nearby damage even if it looks clean. The problem is probably from leakage from the capacitor nearest to the battery. There's a particular via that tends to rot out. Have you checked the advice from this thread? You should check continuity between the two points on the bottom of the board that I jumped together with wire. If there is no continuity, that via probably rotted out. If there is, it might be wise to check all of the other vias in that area of the logic board. You can see the traces underneath U7 and U8 in my picture. Those chips are involved in the startup circuit on the logic board, so broken connections in that part of the board could definitely be causing the problem you're seeing.