I like the Nun's Priest's tale, because it is a fable. He took the qualities of man and exaggerated them into a bird. The obvious moral lesson of the foolishness of succumbing to empty flattery diverts attention from a more subtle warning to beware the advice of women. This was a common theme in Medieval times. They were building on the tale of Adam and Eve, and in this time, they believed (as some still do) that Eve was the temptress and that it was she that caused the fall of man. That is subtly put into this story. However, as the teller is a Priest, you can see why he may do that, and how his interaction with women may be limited. This theme is in deliberate stark contrast to the Tale of Melibeus, whose central figure, Prudence the wise wife, counsels patience and prevents a war.
I much prefer to see the story as one about the dangers of vanity, as the more important and obvious moral.