Just for extra clarity.
The power socket connects the batery +ve ehwn poer is unplugged. Disconnects +ve when plugged in so you dont “charge” the normal battery.
Plgging in the input jack connects the battery -ve so it doesnt drain your bettery when not in use.
I know this is an old post but I had the same issue and finally got round to trying to fix it.
It seems the power socket has a switch that disconnects the battery when power is plugged in but SHOULD reconnect the battery if power is not plugged in.
It then turns on the battery only when the INPUT jack is plugeed in.
On mine, the power socket switch is NOT reconnecting the battery. If I jump a wire from bettery +ve to where it should go, then it will work.
So have ordered a new socket (search for 2.1mm pcb mount guitar pedal supply socket on ebay).
Should be an easy fix as its just 3 legs