What genre or base-game would you want to see redone with Amiibo compatibility as the central mechanic? I am sure many of us immediately think '3D World' because of its magnificant gameplay and its already flirting with giving different powers to different characters (who all have relevant Amiibo, sorta, if you're not a racist Toad).

Some ideas that occur to me in passing include a shooter; kids and teens are all over shooters these days, and most grown gamers could be drawn in by a well made competitve game, with Amiibo's inherent RPG supporting elements, that is drawing inspiration from their own childhoods. Ofcourse depending on style it may run too great a risk of harming Splatoon. 

What genres could you envision? RPG? RTS? Soccer-Sim??

Maybe a RPG sand-box game, with a giant world to explore. Every Amiibo added opens a new region to discover and travel too, with its own missions, rewards and playable characters.

To be honest an Amiibo game running on Mario 3D World engine with perhaps a more action oriented dynamic would be money in the bank. 

Pure bliss.

The engine already kinda accounts for floating characters and characters who shoot projectiles (fire flower) and 4-player local play.

That would be a big, big (biiiiiiiig) hit. 

How giant? I struggle to imagine Xenoblade X as a world size for it. Though have continent sized areas for every franchise covered by Amiibo...

They don't need to be gigantic areas: Maybe you discover a zone the size of a city on GTA III for every franchise you unlock (all Mario themed amiibos unlock the same area, per example).

That could be pretty cool. Any idea what RPG you'd want it to be comparable to in terms of gameplay?

Super Amiibo RPG

Maybe something like the Lego games. Maybe adding elements of other games depending on the area (Turn base fights in the Fire Emblem or Pokemon area, platform in Mario or DK area, racing on Captain Falcon or Sonic...).

To be honest, I feel like that would contradict Nintendo's design philosophy as I've come to understand it. - Usually don't they prefer to expand out from one singular base formula per game?

Yes, but I assume if someone buys an Amiibo is because they like the franchise it's based on. Thus, they would preffer a similar gameplay style.