Riot won't do this because they want game clarity. Honestly, it sort of adds an unnecessary mental factor for people to not get tripped up by looking at the mini map and momentarily not noticing who the champion they are playing against is, which can influence decision making in a negative way. You could argue that that's just another layer of skill in this game, but I don't really think that is the case when recognizing your opponents has nothing to do with champion mechanics, in-game knowledge, and the quality of a leadership role.
I don't personally think I would have a problem with it, but I'm not everyone. I don't think this should be implemented just to keep the game more fair, as cool as it would be to have a marquis of your favorite skin for everyone else to see on the map. Besides, some of the skins would just look terrible as icons. The proportions of their bodies to their environment might make for some awkwardly cropped icons.
If anything, it would be best to have it as a personal aesthetic for you as a player, but I think that's a lot of money Riot would have to invest in player personalization instead of focusing on more dire aspects of the game, such as game balance and fixing client/ server issues.