It'd be an interesting idea. Couple issues though:
1. Rune pages - If people have different rune pages for different champs, they're screwed when the champ changes. And if they have a single rune page for say, and ad carry and an ap carry, then if the champ switches from an ad carry to and ap carry, theyre screwed there too.
2. Item Builds - Different champs have different builds, so you'd have to change your item build almost every time the champ changes, wasting precious time and gold. A possible way to remedy this would be to have the option to buy into multiple different builds, one for each champ you'll be playing with.