Here's why Charizard IS NOT a dragon:
Charizard evolves from Charmeleon, which itself evolves from Charmander. Basically: Salamander, Chameleon, Lizard Char.
Now, what is Charmander? It's a salamander. Salamanders are amphibians - the irony here is that while Charmander is amphibious, it is weak to water! The very element that separates (in the most tertiary of ways) amphibians from lizards. I seem to recall a myth from ancient Europe that says that salamanders are born in flame; obviously burning logs and fleeing salamanders.
So you see, the very basis of Charmander is on an amphibious creature colored by myth. That it happens to grow horns, wings, and look much like a common western European dragon is coincidence; though not by design. Perhaps in the early stages of development Charizard WAS dragon/flying before typing was hammered out, but supposition has no place here.
Now take Gyarados. Why is it water/flying rather than water/dragon? Gyarados LOOKS like a traditional Eastern dragon. But what does it evolve from? A myth-colored goldfish - much like our fire salamander.
Now let's talk Dratini - why is it dragon? It looks like a worm, as does Dragonair, and then you have Dragonite the Orange Fairy Dinosaur. Well it LOOKS fairy-ish, why isn't it fairy subtype? You got me there, but I subscribe to the notion that Fairy typing applies to humanoid or otherwise strange, not-quite-normal "normal" types. Rattatta is fairly normal compared to something like Clefairy, after all. We can go on to argue why Chansey and Blissey aren't fairy-types, but I'm not going into that.
And then there's Garchomp - it's a walking shark! How the hell is it dragon? Salamence looks like a dragon as much as Charizard does, and evolves from a shell that evolves from the mini-me of T-rexes! What about Tyrantrum, which is basically a T-rex? I could go on forever!
Hopefully you get my point - just because it is shaped like a dragon does not necessarily MAKE it a dragon.
Adding a type unnecessarily may also lead to more disadvantages than advantages, and Blastoise doesn't learn any steel moves so why give it steel sub?