I think the difficulty is that a lot of Pokemon that already exist belong to two types that would fall under a single category in your system. What happens to Shedinja? It doesn’t make any sense for him to be not a Bug-type, or not a Ghost-type; the concept just doesn’t allow it. Do you keep Grass an an elemental type, or use Plant as a biological one? You’re in trouble with either Sawsbuck or Abomasnow. Psychic causes similar problems. And what about Pokémon that just plain don’t have any elemental powers, like Raticate?
Besides, the egg group system already provides something like this. It doesn’t play into strategy at all, but why should it need to? Why should a Pokémon’s strengths and weaknesses be different depending on whether it’s a mammal or a reptile?
I think that the biggest objection, though, is that it says “no” in advance to a lot of possible designs for no real reason. Why can’t we have a Pokémon with both ice and electrical powers? Why can’t we have a Pokémon that’s a fusion of a mammal and a bird (gryphons, anyone?)? What reason is there to deny designers that option?