> I had gathered [...] that "Pythagorean intonation (pure fourths and fifths)" WAS the original
> Western Music temperament, WAS the one used in 'Church Modes', and WAS the one called
> 'Just'!!!

You gathered wrong. In just intonation the frequency ratio of a major third (say C to E) is 5:4.
In the Pythagorean system, all notes are derived from successive fifths.   So to get from C to E you have to do that four times, and the ratio you end up with is 81:64, which sounds pretty horrible if you have to listen to it for any length of time. In the Middle Ages that didn't matter since a major third was considered as a dissonance that needed resolving anyway, what really did matter was getting fourths and fifths to sound good. Mediaeval harpists always tune that way. (English musicians may have been an exception, they got into thirds much earlier than anybody else, but they didn't leave any guidelines about how to play their stuff).

By the Baroque, thirds and sixths were considered consonant and bare fourths and fifths were unusual, so musicians used various "meantone" systems that made thirds sound close to just intonation and made fifths sound rather a mess. These systems were often fixed in hardware, as with pipe organs or flutes.

In between, Renaissance musicians tried a mind-boggling range of alternatives, so many that it is hardly ever possible to be sure what the "right" tuning system for any piece really is.