Lemon, take a look at the picture you edited. The "top ground" is term I used to refer to the two ground connections on the right side of the photo.
Anyway, about the blinking LED: From what I can see, it's blinking correctly. The PSU seems to be switching on and off.
Did some measurements with a multimeter and here are the results (I used component markings from TomC's reverse engineered PSU schematic):
When the PSU is on, scope is off and battery is disconnected I get 7.5 V on the D9, EC3, R11 joint and I get 8.5 V on the battery jack and LM324 power pin. Green LED is shining.
When the PSU is on, scope is on and battery is disconnected I get 8.4 V on the D9, EC3, R11 joint and I get 8.3 V on the battery jack and LM324 power pin. Green LED is shining.
Rail marked -7.6 V reads -7.55 V on the adapter board power cable and voltage marked 8.4 V(sw) is 7.6 V to 7.7 V
When the PSU is on, scope is off and battery is connected I get 1.5 V on the D9, EC3, R11 joint, jumping up when LED blinks and I get 7.73 V on the battery jack and LM324 power pin. Green and red LED is blinking.
When the PSU is on, scope is on and battery is connected I get 1.5 V on the D9, EC3, R11 joint, jumping up when LED blinks and I get 7.4 V on the battery jack and LM324 power pin. Green and red LED is blinking.
Rail marked -7.6 V reads -7.55 V on the adapter board power cable and voltage marked 8.4 V(sw) is 6.7 V to 6.8 V, jumping up when the LED blinks. I can also hear a tick whenever the LED blinks. I guess this is probably from the movement of the coils.
I believe that when the power is on and the battery is in, the scope is actually running from the battery and not from the mains power. PSU also seems to be resetting all the time for some reason and will not enter current limiting mode to charge the battery. It's behaving as if the output is shorted. I'm currently charging the battery by hand and testing out how the PSU behaves with different charge levels. It's voltage is at 8.17 V now and there are no changes to the way PSU is responding. I'll do some more checking on the first tho op-amps on the LM324. To me it seems as if the problem could be somewhere close to them.