The connector going to the actuator has three wires
0-12 volts[the output of the temp pot]
an easy first step is to find the white connector to the pigtail going down to the servo and check those voltages.
I have not done this, but theoretically you could:
supply the servo pigtail ground and 12 volts and then 12 volts to the variable wire, which with a working servo should either open or close the flap. Then 12 volts and ground, then ground to the variable wire should do the opposite to the flap. This is only with the pigtail detached from the rest of the car[else magic smoke gets out]
IIRC the servo is held in with 3x 4mm screws. If one were to remove the top clamp on the heater box, which is the only thing keeping it from rattling around, I wonder if one could shove it around a bit, and with small enough fingers and wrenches one could loosen them. There is almost no way in hell to get them back in, but then the flap would be loose and one could seasonally flip it until the whole HVAC rehab is in the cards
In the OP's pic the little white connector hiding under the heater hose should be the servo connector, and the pigtail disappears under the little plastic cover circled by cyow5