Similarly, I haven't seen the illustration of "Where does this first go from a true equation to a false equation?" used as a way of teaching the underlying concept,

My middle school algebra books discussed this on a very basic level. The problem in the book would show the work of a fictional child who was trying to solve a math problem and failing, and the real student would be told to identify where the first one went wrong and how the problem should have been solved instead.

I never saw it done with anything more complicated than algebra, though.

I'm not sure whether or not this is common, either.