Well, here is where it gets interesting.
The next best repair shop guy's idea was as suggested here, press on the remaining bit that the top broke off of, then push & pull the whole thing.
He broke the plug within 5 seconds, then broke the "handle" on the black, middle plug. At this point he was unwilling to continue and sent me on my way. ?
Interestingly, even though the whole inner "wall" that goes around the hook is gone, it still stuck on anyways, as it appears that the non-round corner of the plug does some magic gripping that keeps it on.
After some wiggling, I got the small plug off as well with the help of the tool shown in #27.
Finally, for the middle, black one, after loosening an insert piece at the top of the plug, I was able to press that one down and fit a screwdriver in the small hole to bend the inner wall out and get that one off.
Next, I got to change out my control unit for a used part I'd gotten off ebay weeks ago.
Reseat the plugs and.... nothing
Looks like the whole "fan turns on when hammering on the control module" thing was random and something else is broken. Good times.
3dB even suggested that it might be the fan itself
Do not blame the fan control module before you checked out fan itself. It has brushes that can get corroded and disconnected or plainly worn out. I had same problem with my 1.4TDI - fan not working, temps climbing. At that time had not any spare money available to buy a new fan, so I fixed existing one. One brush was plain stuck and other had corroded connection. Test is simple - try to hook fan up directly to 12V and see if it works. It should spin pretty fast on full power.
but imo, the facts that I've seen it turn by itself and that it is easily movable by hand tells me that might not be it.
How does one find & fix something that makes no effort to tell you it's broken? ?
Going to be fun tomorrow in 30°C + heater on I guess