Well first your making the assumption that the carb is actually adjusted correctly (it probably is in your case, as your driving it). You would be amazed how many owners of Treinkel never look into their carbs to see if the adjustment of the cable and compensator is correct, and they have full throttle movement. It is rather satisfying to adjust the throttle slide so it actually opens the last third, and watch them be stunned at the sudden additional performance for about two minutes fiddling. Of course then you have to tune old carby in again, but that's another story.
So I bet there are cars that have nigh on closed, or open, carbs for tickover running the wrong mix. Given that is the case advice can be incorrect as is the car overall weak or rich? Can be both, if its well out, at differing openings. The thing is not idiot proof.
I thought the cut out in the front of some slides was to proportionally increase the air flow for small movements near tick over. In other words the volume of air remains low but the majority of the air passes directly over the needle aperture drawing out more fuel then the equivalent none cut out airflow where much of the air does not pass over the needle aperture. Thus in such a position the ratio of fuel is a little richer than when a full throttle, if all other things are stable. The needle is, of course, shaped as a taper. That taper controls the mix when the needle is withdrawn compensating for the lack of control from the front of the slide. So the one takes over from the other, meaning the ratio can be the same, if desired, in fully open positions. In practice it probably is not. Go further and in SU carbs tuning should always refer back to which needle to use in the carb to match the rest of the engines performance. A standard needle wit5h a high lift cam, for instance, is a waste of time. You need a needle to match the changed spec.
Certainly on these strokers, Sachs and Goggo, I am in the habit of giving them a tweak of choke down hill if not under power. Saab, Trabbi and such do not need this as they have freewheel to protect them. On both those smaller cars the choke is little more than a tickler. Tickling and choke are actually performing slightly differing functions. Your hand can mimic a choke, so starting cars as if by magic, but it cannot mimic a tickler.