I think having a free downloadable game, that is updated constantly, as a means to build up your character and perhaps find secrets to use in other Nintendo games that support Amiibo is a great idea.
However, I would not make it a dungeon crawler. Whatever the game is, must be able to support all the characters unique abilities and attributes. I am thinking a 3D platformer with some borrowed gameplay from Zelda and Jet Force would allow all characters to shine.
Then Nintendo allows you to find bonus items for other games. Like building a new Kart for Mario Kart 8, or finding exclusive weapon upgrades for Zelda...something that makes the game easier, but not necessary to win the game.
Another idea I would like Nintendo to explore is the Super Mario Bros. Crossover fan game.
Imagine having a Nintendo Crossover game. You buy a character and get a few levels from that characters game you can play. But get several Amiibos and now you can play your characters in all those games...allowing for quite a unique game experience.