Yeah, pokemon can beat their limitations... by training with their trainer. That's what EVs are. IVs are how they are born. Sorry but you won't get that argument across with me. IV training literally redefines what IVs are. It's ok if you like it: but call it what it is. Redifining what IVs are.
EVs are "Effort Values". It's what you achieve by regular training, your specialty. That's why starts at 0. By getting it to 252, a Pokémon is not "exceeding its limitation", it's specializing in that stat.
IVs, "Individual Values", are something the Pokémon is born with, defining if it's naturally faster, stronger or bulkier. It's kinda like it's genetic code. It's a "personal", even "biological" limitation, since one is born and dies with it intact.
Yeah, it made sense for IVs to be permanent, but it also makes plenty of sense for a Pokémon, along with a Trainer it trusts, and after achieving what used to be its "maximum" (level 100), to surpass those "genetic" limits through a special training.
Maybe it doesn't make sense for a "real world logic", but it's perfectly fine for a "Pokémon logic".
If something about all those mechanics makes absolutely no sense is resetting EVs, since it's like "forgetting" or "unlearning", specially making it through eating Berries and punching bags. But hey, it's a game. Remember the 4-moves limitation, that also makes no sense, and chill.
I'd even go as far as to make Hyper Training accessible before level 100, maybe by maxing friendship, or Refresh, just for the sake of convenience.
Damn, for that matter, we could even manipulate Hidden Power through a similar method. We'd give the Pokémon along with some special items (shards, bottle caps, whatever), and come back after maybe 1 day or so, and the NPC would have trained it to learn a new Hidden Power.
Now, if we were talking about manipulating natures, that really would sound a stretch. Weird, actually. And it's not really necessary, since it's the most easily achieved of all aspects of a Pokémon. But again, it's a game, I wouldn't be entirely surprised if GF decided to make it manipulable.
EDIT: The idea of changing the benefited and hindered stats through a Psychic NPC solves the problem. I'm all in for that!
For a HA capsule, it would be cool to have it, but being extremely rare. Like, you get only one in-game, some legendaries/mythicals holding it on certain give-aways, maybe as a prize for participating on the tournaments, and so on. Something rare enough to only be used on those special shiny Pokémons that happened to come with the wrong ability.