Hello,
'To me, it sounds like there's either a huge amount of play in the steering (which would explain the need for constant counter-steering), or a complete failure of the power steering system (which would explain the 'strong pulling' sensation on the steering wheel).'
But a quarter turn of counter-steering?

That would mean that the top point of the steering wheel would be the rightmost point when counter-steering (rotated 90 degrees). That sounds like so much that the error must be obvious to an experienced eye.
Okay, let's jack up the front of the car and check the wheel play. Put one hand on the front of the wheel and the other on the back, and rotate the wheel back and forth. There shouldn't be much play at all.
In addition, I would like you to check the play in all the rubber bushings by inserting a long lever between them and seeing if there is any movement.
Does the servo make noise when the car is stationary? Could the V-belt be slipping? Please perform a visual inspection of the V-belt. Also, with the engine running, try turning the steering wheel fully to the right and then to the left. The V-belt should not squeal or slip during this process.
I had a similar effect with my previous car (a Scorpio). Whenever it rained, water would splash onto the ribbed belt, causing it to slip.

(This was all because a splash guard was missing.) This also resulted in the car strongly following the ruts in the road and having difficulty getting out of them.
If the servo actually fails, I would suspect the pump rather than the steering system itself. I have never heard of a servo failure without some kind of mechanical force (i.e., an accident) being involved.
Regards,
Private.