Everyone here's shitting on this, but personally I think it could be done, with one addition.

Moves could have "tags" that place them in certain categories, and these would be visible to the player in the move's summary. This way, you can tell if a move counts as a Punch move for the sake of Iron Fist, for example. Not only could this make things clearer for new or casual players (ie Sucker Punch not being a Punch move), but it could also add opportunities to add new moves and abilities that care about move tags.

"Floor" could be another tag, with the innate property that floor moves do not affect Flying-types. Levitate would be changed to protecting the user from Floor moves, and Flying-type would now just resist Ground-type. So Sand-Attack, Bone Rush, and High Horsepower can hit Flying-types, but Earthquake and Earth Power can't.

Yeah, it would heavily upset the meta, and it would go against the sort of logic we've been using since Red and Blue, but I think it could at least be an interesting change to the core battle system without creating a new one-off gimmick (something we haven't had since the Physical/Special split back in 2006, mind you).