No, I do not think Jane Austen does not privilege education over nature in her novel Pride and Prejudice. Consider Miss Bingley: she is highly educated (for a woman of this time), speaks several languages, and plays multiple instruments. She is quite worldly. Elizabeth, on the other hand, is not well educated. Her parents put little effort into schooling their children, and they never hired a governess to attend to the girls' education. Consequently, the Bennet sisters were left to engage themselves as they chose: Elizabeth reads quite a bit and is very intelligent, but she doesn't have the same ability at the piano or with languages as Miss Bingley.
If Austen wanted to compel readers to believe education is the key to being truly well-rounded, likeable, or even good, then she would make it much easier for us to like Miss Bingley. Miss Bingley, however, is a snob who would rather put on shows to attract Mr. Darcy than spend time to improve herself. Elizabeth is much more likeable, and she follows her own nature. When her mother insists she cannot walk all the way to Netherfield, she does it anyway. When Mr. Bingley's sisters talk about Elizabeth's dirty skirts behind her back, Mr. Darcy defends Elizabeth's concern for her sister and says the walk seems to have enhanced Elizabeth's beauty. Characters who are good and discerning prefer Elizabeth; Darcy, Bingley, and Jane all think highly of her, and Austen makes it easy (in the end) for us to like all of them. This is how we know Austen ultimately privileges nature over education; if Elizabeth represents nature, and she is honest, upright, smart, and kind, then we can know nature is privileged over education, which is represented by characters like Miss Bingley.