Regardless of being 'proper representation' or not, which is a subjective opinion, the ways in which Pokemon is represented defy how any other series is represented in Smash Bros. Thus far, this is all fact.
Many believe random assortments of popular Pokemon due the trick- but I suggest this claim is ridiculous, that it is in fact possible to represent the franchise, despite its large number of characters, by focusing on what makes the series what it happens to be.
We absolutely cannot add everyones favorite Pokemon, or even most of them, so that goal is immediately futile.
Instead, we could focus on making it feel as though the world of Pokemon is in smash bros, not just the characters. I postulate this is an achievable goal, and that it cannot be done just by adding the 'popular Pokemon' and calling it a day.
First off, Jigglypuff is supposed to be seen as a balloon. It is called the "Balloon Pokemon" after all. I guess they may not have hinted that very well in anime, but I think its move set makes sense. Heck, it even has Sing, which Jigglypuff is known for. I think they represented Jigglypuff quite well.
Jigglypuff, species: Balloon. Mewtwo, species: Genetic. Charizard, species: Flame. Ivysaur, species: seed. Greninja, species: ninja.
You're telling me it only matters for two? The species of Pokemon doesn't make any sense anyway, but if the only canonical ability of Sakurai is to take ninja and balloon, then the others should have been represented equally. Mewtwo's species is not even a noun.
And by the by: Pokemon is not the only franchise where only new and relevant characters are added. Fire Emblem suffers from that problem too. Yes, we ended up getting Roy, a classic character for DLC, but he ended up getting in the game due to fan demand just like Mewtwo.The only franchises in Smash I know that get characters regardless of relevancy are Kirby, Mario and Zelda, mostly because they have a lot of reoccurring and popular characters.
It is the only franchise that only, solely, cares about popularity. Fire Emblem adds more than just new and relevant characters- they add the main characters of the games- the only case where they arguably didn't is Robin, and this was stated to add the 'essence' of the series, which the Pokemon alone do not do for it. I'm not sure where you get relevancy from, you brought up the point. There are plenty of characters relevant to Pokemon, that are not simply popular. All the ones we get are just a hodgepodge of popularity.
The reason why Pokemon is the only game in Smash that does not have the "protagonist" playable, is because the protagonist is actually not that important to the Pokemon series. The Pokemon Trainer serves as an avatar for the player, and while he is kind of important to the Pokemon universe, he is ultimately not as important as the Pokemon. I mean, the game is titled Pokemon, not Pokemon and Humans. Sakurai did like the Pokemon Trainer idea at some point; if he didn't, then why did he add him in Brawl? It turns out though, he didn't find the idea as amazingly executed as he thought it was, so he scrapped him for Smash 4.
Pokemon may be the name of the franchise, but that doesn't mean that all thats important are the Pokemon. Zelda isn't the prominent character in her series. Even more so, the closest game in regard to Pokemon (hundreds of characters, one main avatar) is Animal Crossing. That is a game named after the other characters of whom you do not play as, yet the remedy for that series is to JUST add the protagonist.
Besides that, yes... the Trainer is the vehicle of the entire story. Pokemon are the point of the franchise- and that is why we have tons of spinoffs without a trainer. Yet, in the base game, the trainer is CRUCIAL, and the mechanics of catching and training Pokemon, strategizing a team, are all about the trainer. The journey the player goes through is as an actual person, personified in the avatar trainer. Most of the Pokemon are even marginalized- captured and stored in a box forever to be 'completed,' and otherwise disregarded in their game.
The pokemon are very important to the game, absolutely, but the Trainer is a crucial part to how the game plays. Pokemon without the trainer is Mystery Dungeon. That's an entirely different game.
Don't get me wrong; I think the Pokemon Trainer is important enough to the Pokemon series to be added in Smash. Again, even though the Pokemon Trainer is not as important as the Pokemon, they are important enough to receive some sort of recognition. However, Sakurai tried to incorporate him; it didn't work out, so it's just something that we have to deal with. :/ At least there are several Pokemon Trainer trophies in Smash, like the X and Y trainers and Prof. Sycamore. It's better than nothing, I would say.
Also, what do you mean by Sakurai "making it up?" You do know that Sakurai has "made up" move sets before, right? He made up Wii Fit Trainer, Captain Falcon and Fox completely from scratch, despite the characters not having any sort of hand-to-hand combat in any of their games. Does that mean they're not represented well by your logic? I'm curious.
He made up Captain Falcon and Fox... Wii Fit trainer does yoga, which the character coaches you through. Fox uses weapons that his ship does. Falcon is the only one you listed whom he actually made up a moveset for moreso than any other character. My point was that Greninja was not based on the games, it was based on a sketch. That's it. The character we have in the game is based on a sketch. This is inherently, explicitly, a non-attempt to actually represent the character. He said himself (and only for Greninja, as far as I'm aware, no one else), that he based a moveset on a picture. If you think this is 'properly representing the character,' then be my guest to back it up.