It would be nice if you did your own homework instead of claiming immunity because you gave your texts and journals away. You can haul yourself to a library as easily as any one of us. Since you're retired, you might be able to do it *more* easily! Instead of complaining that a mode or median is not an average, you might look at where the notion came from. With your knowledge of statistics, you're in a better position than most to sort it out.
I'm starting to see where "median (and possibly mode) as an average" came from. Back in 1874, Galton was looking to investigate inheritance of difference among people, his "statistics by intercomparison". It involved looking at "deviations from an average". In some instances the "average" that he felt it was best to consider was the median. It may well be that the practice predates Galton. I'll let someone else trace it back farther.
I'm done with this discussion because I'm tired of doing all the work. You've now got Galton, Yule, Kendall, Snedecor, Cochran, Wallis, Roberts all using average as a collective. The concept of "deviations from an average" seem like an excellent way for this notion to have started. I don't have the time or interest to pursue it further.
The historical use of average as a collective, which now has been demonstrated to span over 100 years(!), is why I tell my students not to use "average" but instead say whether they are talking about "mean", "median", or "mode". I don't tell them to ignore 100 years of usage by some of the leading people in the field.