Comment by honzabe
> all it took was a single run with someone holding the seat
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".
Grabbing the handlebars is the absolute worst way to try to teach someone, no wonder the author had problems. I can't believe that this is "the usual manner" either, or at least I hope so (for the sake of the children)