(Although I should think the profile would drive the media rather than vice-versa. A media can support many kinds of papers, but a paper only recommends a specific media.)
It's a 1:1 relationship no matter how you do it so it makes more sense to select what printer you are using, then there will be a dozen or so media types for that printer, and that dictates the profile. Then you can drill down from the printer to the media rather than searching through potentially hundreds of random profiles for random printers. Just makes more sense to start at the top rather than down in the weeds.
Mike