I think this is the key - holding the seat. The author of the article says "I did so in the usual manner - have her sit on the seat while I grab the handlebars and run along side her" - this surprised me, it would never occur to me to grab handlebars. I am not sure why exactly, it's an intuitive thing. As a lifelong road cyclist and an ex-racer, I have a pretty good feel for a bike... and you don't actually steer with handlebars, you "steer" (as in 'control where the bike is going') with your ass (which is why every experienced cyclist can easily turn with both hands off the handlebar). Someone grabbing your handlebars would misalign where your body is going vs where your front wheel is going and this makes you fall.
EDIT: I just realized that "steer with your ass" is something our coach used to say when we were kids. I think "your ass" is a kid-friendly proxy for "your center of mass".
Also the people that are 'good with bikes' are the absolute dog shit at teaching other people how to do it. In fact I consider every advice from them with the maximum caution possible, because it is, a lot of times, actually harmful
As for the "teacher".. I forgot to mention that: The kids did it all by themselves. One of the older kids, i.e. 7 or 8, would hold the seat and run behind/besides the kid learning to bike. The trick was clearly to stay far enough behind so that the kid learning didn't notice when the "teacher" let go of the seat.