... I am extremely skeptical that you can take a non-swimmer and have them swimming "well" in a matter of minutes....
I, too, am skeptical. I learned to swim as a young child (my mom insisted, for which I am eternally grateful) and have always been an okay swimmer. A couple of decades ago, as an adult, I decided to improve my form by taking a swim class at the local university. This was a one-semester class that taught five basic strokes. Learning to swim five basic strokes with reasonable efficiency took the whole semester.
Fifteen to twenty minutes might be enough to demonstrate all aspects of proper form in one stroke (say, the crawl, which I would say is the most basic stroke) but that's only the first baby step toward teaching someone to swim. The student then needs to learn how to make their muscles do what they've been shown, which sounds simple but is not a trivial matter, then needs to practice, with constant corrective feedback, until muscle memory makes the movement instinctual. And then, to be a strong swimmer, the student needs to build up muscle strength by regular swimming over a period of time, though this last need not be in class.