See, the problem with Dragon isn't just that it can cover its weaknesses. I agree with you that that can be good strategy. But when Dragon is populated by many legendaries and pseudo legends that combine that coverage with ridiculous BSTs and such so that they outspeed and/or overpower you before you even have the chance to counter them, it becomes a major problem.
Aside from that, I think another type chart change is warranted, there's a few lingering issues in the current type chart:
-Electric is only weak to Ground. This is a definite problem, but not for the reason you expect. Electric may only have one weakness, but it only has decent coverage, it's not as if the type itself is OP. The problem is that this gives Ground a near monopoly on Electric type coverage and almost forces you to have one on your team (sure, you could have a Grass or Dragon type or something and not get the SE, but it's usually better to have that advantage).
-Ground itself is pretty OP. It's SE against 5 types, and only 3 types resist/are immune to it. And with Earthquake being a TM move, there's very little to discourage this type from running rampant. It could definitely stand to be brought down a peg.
-Ice only resists itself, making it worthless defensively (and sadly, many Ice types have BSTs which are more defense oriented).
-Dark and Ghost share the same type advantages, making them fairly interchangeable and not very unique.
-Less of a priority, but anything that can be done to help Grass, Bug, or Rock would be great. They all have tons of weaknesses (which is especially damaging for Rock, which is conceptually designed as a defense oriented type), and half the type chart resists them. Pretty much as long as they don't exacerbate these issues like Fairy did for Bug, that would work, but giving them new type advantages or having them resist new types would be ideal.
I think they could fix it by doing this:
-Have Ice resist Ground and Dragon.
-Remove Fairy's resistance to Bug.
-Add two new types to handle the problems with Electric, Ghost, and Dark. Not really sure what type concepts could fulfill these requirements, but we'd need at least two of them to fix the Dark/Ghost issue (otherwise either they end up giving both of them a type advantage to the new type which just maintains the problem with them having the same type advantages, or they just give it to one and the other becomes less useful). I've heard Alien type thrown around before and that would certainly fix Electric (since it would make sense for Alien to be SE against Electric since aliens could disrupt electricity) but I'm not as sure it could help fix the Dark/Ghost issue (could Ghost be SE against it? I'm not sure that'd make sense). Aside from that, I could also see it being given type advantages and weaknesses that might help the balance even further, like being SE against Normal and weak to Poison.