Thanks to Dave L. for alerting me to this very erudite discussion!
Re. marketing material; there is no strong evidence that any of the "set play" sequences demonstrated by Barton-Wright and Pierre Vigny for B-W's Pearson's Magazine article were performed verbatim during training at the original Bartitsu Club in London. On that basis, it's arguable that they may have simply been improvising for the camera; the articles themselves definitely doubled as marketing for the school. However, another article written by Captain Laing, a soldier who had trained at the Bartitsu Club for three months, recorded a number of similar "set play" sequences, which serves as evidence that the Club did employ this type of training, if not necessarily the specific sequences shown in the Pearson's articles.
Re. the practicality or otherwise of the Pearson's set-plays; the overall premise of the current revival is that Bartitsu was essentially an experiment in cross-training, abandoned as a work-in-progress in 1902. The object of the revival is to try to pick up where Barton-Wright left off. The revival is a very "open source" movement and every instructor works out their own relationship to the original (or "canonical") material, including the walking stick articles. My own take is to work with the canonical set-plays at three levels:
First, they are practiced as a mark of respect and out of purely historical, academic interest; it's fun and, in a "living history" sense, valuable to be able to learn the sequences exactly as Barton-Wright demonstrated them. At that level, they also serve as a useful common technical and tactical "language" for the revival movement, directly comparable to ko-ryu kata, etc.
Second, practiced verbatim they teach some generally useful skills of combat body mechanics; extension, alignment, tactile response, etc.
Third, the canonical set-plays really come to life when you mess them up. A central exercise in my own Bartitsu classes is to allow the "opponent" to spontaneously defeat the "defender's" pre-arranged responses, forcing the defender to improvise solutions. There's a good example of this semi-freestyle drill applied to the technique in question at 2:18 in this video clip, which was also shot at the 2010 seminar in Eugene, Oregon. Given permission to defeat the set-play, my demo. partner crashed through the guard, pulling me down into an awkward semi-crouch; I improvised a response by converting the crouched position into a foot-sweep and takedown. As mentioned in the video captions (and the whole video is worth watching to get a sense of how this type of drill works in practice), this "combat improv" drill provides a useful "bridge" between the formal set-plays and free sparring.
Regarding the ankle-hook technique specifically; several of the existing answers have already nailed it, especially re. "connecting" the cane to the defender's hips, so that the sudden backward sliding step engages the core muscles and the defender's weight against the opponent's ankle, and also the value of being able to "read" measure (combat distance) and so-on. I would only add that there's also an element of pain compliance when the hard inner surface of a cane crook is applied to the bony ankle.
Finally, although I don't regard this sequence as a "high percentage" move and would not teach it verbatim as self defence per se, I was told that a seminar student actually did successfully execute the "ankle pull" aspect in a real-world self defence situation. Wonders will never cease ...