The one thing about "competitive Pokemon", and what we understand it to be, is that it's something created between people and the community.
While you could create trainers that introduce competitive concepts, whether that's simple strategies like Trick Room or weather, or team composition 101, that is only a basic emulation. You can't readily replicate competitive play with AI and preconstructed teams, because you run into the matter of human intelligence, prediction, variability in teams and attempts to topple competitive trends, and so on and so forth. It just won't work.
Now, that's not to say that the game wouldn't benefit to have some sort of basic introduction to competitive concepts, like with the examples I used.
It could be really useful to create some in-game "crash courses" for significant things such as EVs + IVs + natures and their influence on stats, the ins and outs of breeding a good Pokemon, and the principles of putting a Pokemon together (e.g. coordinating moveset, stats and item). That could help introduce curious people to the hellscape that is competitive, without them having to go look up how EVs and IVs work, how breeding works on here, and so on and so forth.
It was already said that PVP is the PokeMMO endgame, and ultimately that's how every (mainstream) Pokemon game tends to go: you clear the story mode, finish your Pokedex, and... then what? If you want to keep getting a fulfilling experience, you start deviating towards breeding, RNG manipulation, PVP, shiny hunting, all of those little things that play to our perfectionist and competitive impulses.
There's going to have to be content that supports that endgame here, because that's where it'll end up for everyone eventually.
Besides, it seems incredibly unfair for there to be no in-game "crash courses" for this content, when that content has already been implemented by means of the Elite 4 existing in PVE. A well-constructed competitive team will always be able to crush a casual team: so why set players up for frustration if they don't know the fine details?
Actually, this is a fair point too. As much as I love difficulty that requires creative play-arounds, there's a line to be drawn between "creative solutions are needed" and "just plain unfair". The occasional hold item and obscure, weird or rare Pokemon make the player have to think outside the box, which is perfectly healthy. But you want to keep in mind the resources that the player is given vs. the resources used by, and applied to enemy trainers.. and find a fine balance between them.
tl;dr Competitive as we know it is a social construct, so it's hard to emulate in the solo experience; however, having ways to introduce competitive concepts and how to build good Pokemon + teams wouldn't hurt whatsoever, because "competitive" is almost always going to be the endgame for players on here.