Not really, no.  I actually felt like we had enough to cover most any design even before Fairy was added, and I tend to prefer parsimony in these things.  The only thing I felt could be missing was something along the lines of a ‘Holy’ type, and Fairy is probably close enough to cover most designs that would fall within that.  If anything I feel like the game would survive with fewer types, not more (do we really need Ground and Normal?).

Grass… kinda gets a raw deal, yeah.  Just purely in terms of the number of other types it’s strong/weak against, it’s one of the games’ worst… which is kind of a sore spot for me, since it’s also my favourite.  Most types are defensively vulnerable to two or three others; a few are vulnerable to only one, and Ice (which also kinda got shafted, I think mainly because Generations 1 and 2 had no pure Ice-types) is vulnerable to four.  Grass is vulnerable to five: Flying, Ice, Fire, Bug and Poison.  Only one other type has that many defensive weaknesses – ironically, that would be Rock (vulnerable to Fighting, Steel, Ground, Grass and Water).  Rock, though, gets to enjoy being one of the game’s better offensive types – Rock is the only type other than Ground to be strong against a greater number of types than it is weak against, offensively.  Grass, not so much – Grass attacks are weak against seven different types (Grass, Flying, Bug, Fire, Dragon, Poison and Steel), which until X and Y was more than any other – and that changed not because Grass’s situation improved, but because the next-worst-off type, Bug, gained a new disadvantage against Fairy-types.  They’re strong against three (Rock, Ground and Water), which is sort of average, really.

Grass does, admittedly, have a decent number of resistances – four of them (Water, Electric, Ground and Grass itself), while many types only have two or three, but it doesn’t really stand out in that regard – Water, Rock and Dragon also have four, as do Ghost, Fairy and Flying if you count their immunities (which, y’know, are kind of better), Poison has five, Fire has six, and Steel blows them all out of the water with eleven.  It has to be said that counting types can only go so far – Dragon, for instance, was hands-down the best attacking type of the fourth and fifth generations, despite being strong against only one type (itself), because it had nearly perfect neutral coverage, resisted only by Steel.  The problem, though, is that where types like Dragon and Rock lose out in one respect but do very well in another, Grass kinda loses everywhere.  Grass-types are, admittedly, also immune to Leech Seed and, as of X and Y, powder attacks (the important ones being Stun Spore, Sleep Powder and Spore) – so basically they’re really good at blocking other Grass-types.  That seems to be the niche Game Freak has in mind for them here.  I mean… not that those immunities aren’t useful, but they’re also kind of a slap in the face.

So, when I put it like that, it seems like the obvious thing to do is strip out some of those damned weaknesses (the attacks, I think, are fine as they are – Solarbeam could maybe use some work, because at the moment it’s kind of a gimmick and only viable on dedicated sun-abuse teams, but I’m not sure how to change it).  Let’s start with Flying.  Why does Flying beat Grass?  I’ve seen element-based systems before where Plant/Forest is actually strong against Wind/Sky, birds can help plants through pollination (analogy with the Grass-Water relationship), and it’s scientific fact that plants bolster hillsides against erosion by the wind.  I say make Grass-types resist Flying attacks, and Grass attacks neutral to Flying-types.  Make Flying-types strong against Water instead; the smug pricks deserve it.  We can probably get away with removing Steel’s resistance to Grass too, since Steel has too many damn resistances anyway.  In flavour terms that’s not as solid, because there’s precedent for Metal-beats-Wood in the Chinese Wu Xing cycles, but the fact is, there are probably ways to justify having Steel resist everything, so I think a little more restraint is in order when dealing with that type.  Those things won’t make Grass an amazing type, but it’ll certainly make it not suck!

Bug could use some help too, but I’ve been babbling long enough…

Flying is weird, because it seems like it can consist of either being a bird or having the ability to fly – not necessarily both (see: Dodrio, Scyther).  In addition to the fact that there’s only one straight Flying-type (and even that in generation five), it might also be important that Flying is almost never the first of two types; there’s no functional distinction at all, but which element gets put first often seems to say something about which one is considered more important to the design (e.g. Water/Rock – Relicanth and Corsola – vs. Rock/Water – Kabutops and Omastar).  Noibat and Noivern, only just introduced with X and Y, are the only Pokémon to put Flying first.  What’s more, wind powers – which we now associate with Flying-types, and which are Tornadus’ main feature – originally seem to have been connected with Normal, not Flying, because Gust was a Normal-type attack in Red and Blue, while Whirlwind and Razor Wind, the other wind attacks that existed in the original game, still are Normal.  I think what all this adds up to is that Flying wasn’t originally considered a type, as such – it acts more like a trait or ability that certain Pokémon possess, which makes sense when you think about it, because that’s what flight is (Dodrio is explained by the fact that, as we know from Missingno., the type was originally called “Bird,” not “Flying”).  Tornadus shows that Game Freak’s conception of what the type means has changed significantly since Pokémon began, but some things about this game resist change.  Charizard, then, isn’t so much “a Fire/Flying-type” as “a Fire-type who can fly,” while Pidgeot – and this is important when we move into the next half of your question – isn’t “a Normal/Flying-type,” but “a Normal-type who can fly.”

Now, Normal-types.  They rely primarily on attacks of pure bodily strength, but without the complex motions and training that allow Fighting-types to overpower them hand-to-hand.  As such, they also find it difficult to harm the rigid, reinforced hides of Rock- and Steel-types, and cannot inflict any harm at all on the insubstantial Ghost-types.  However, being more aware than most Pokémon of the limitations of physical reality also makes them distant from the world of ghosts and spirits, rendering them invulnerable to harm from that direction.  So, what do I think are the defining attributes of the Normal-type?

None.  It doesn’t have any.

Normal is the “everything else” type, where we find Pokémon who have powers that either relate to none of the other elements (like Pigeotto’s wind powers, Jigglypuff’s song, or Chansey’s healing abilities), or that encompass all of them (like Ditto’s transformation skill).  This becomes difficult when we confront dual-typed Normal Pokémon… but other than Normal/Flying-types, whom we’ve discussed already, how many of those are there in the first three generations?  I count only Girafarig, who is something of a special case because his design focuses on his bifurcated nature; his front half has the more cerebral temperament associated with Psychic-types, while his back half is animalistic and has no special powers.  It’s not until Bibarel in generation four, then Sawsbuck and Meloetta in five, that this really starts to change, and only now, with the introduction of Diggersby, Heliolisk and Pyroar, and the retyping of Jigglypuff, that non-Flying Normal dual-types have become a significant element of Pokémon’s diversity.  What this says about those Pokémon, I think, is that their elemental powers are comparatively less developed than in the case of other Ground-, Electric-, Fire- or Fairy-types, and that they rely as much or more on their non-elemental abilities (sound-related powers are still unaligned, for instance – hence Jigglypuff, Meloetta and Pyroar, who has a sound-based signature move, Noble Roar).  It was difficult to give Normal-types abilities related to types other than Flying for a long time because the thing that made them Normal-types was their lack of any such abilities.  It’s only fairly recently that they’ve been able to reconcile this with their desire to use a greater variety of Normal dual-types.