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I would look at pins 4 and 5 at the battery connector, it looks like a problem with those data lines and it can't detect the battery, It appears the charger and battery line tie together, so maybe when it can't
detect the battery it won't turn on, but once it is on, the battery power is bonded to the power so it will draw from there once the charger says DC is available. Also possibly R6950, which looks is the path for the battery to ground for it to detect it is connected to a computer, maybe if that isn't there it won't talk on 4/5. Just speculating from the schematic.