For Mythicals, I think story and a place in the setting is fine and even to be expected. What distinguishes it for me is how widespread that story is within the universe vs a Legendary. To go over what I think is a good and what I think is a bad example for some comparison (mostly going off main game depictions because Anime Movie stories and loose continuity are a mess alongside Spin-offs)
- Darkrai is a Mythical Pokemon I like for having a clear lore and place for existing within the World, characterized by the fact that the world's people have little idea or frame of reference for it. You the PLAYER understand it with clarity (especially when Arceus lays it out in Platinum's Encounter), but to the characters it's just an enigma on Newmoon Island that they're not even clear is a Pokemon specifically. There is a widely unknown story that the Player/Meta viewer learns but remains just an unwitnessed Myth diegetically.
- On the other hand, I very much dislike something like Diancie which is known by someone to the point they can have people roaming around searching for information about it and evidently not be met with "what the heck are you talking about?" This makes it sound like a standard Legendary in a sense, as it's well known enough to believe is real and search for rather than write off as a tale or a phenomena.
- There's also odd cases like Zeraora or Zarude where they don't really have any major story going for them, known or unknown to the world, and just feel like they're Mythical as a marketing ploy very transparently. These feel weak in a different manner because there isn't really even a "Myth" for the Mythical, and it almost brings into question how arbitrary the "Legendary" status is for anything besides "specifically 1-off/divine" Pokemon like Groudon/Kyogre or Xerneas/Yveltal/Zygarde as opposed to depicted-multiples like the Kanto Birds. At that point what makes them different from "mundane" species with great power or folklore/urban legends of their own like Banette being an animated Doll or Volcarona worshipped as a Sun God?
Pecharunt as our most recent example kind of falls in a weird grey zone for me: the Story it's connected to is a very well known one with the Loyal Three (I wasn't clear if the story TPC shared on Youtube is referenced or known in the games themselves or was just audience context), but Pecharunt itself is so obscure that no one remembers it existing nor connects it with certainty once it does make itself known again.
Ultimately I think Mythicals benefit most from having a clearly defined story or lore that the world itself doesn't/barely knows anything about even on the level of a "probably made-up" dismissal.
I don't agree because Mythicals feel explicitly the opposite.
Legendaries are supposed to have a clear place in the world, Pokemon that are often not seen by most people but have cultural significance. I'd go as far as to say Darkrai showing itself to people ruins its appeal to me. Why should it have a "clear place" in this world? That's the opposite, if you asked me, of what it means to be a Mythical.
Discounting the anime since I don't see it as the same canon, and then we'd also have to go all over the place:
Celebi doesn't have a "place in this world", neither did Mew, other than being used by Team Rocket to create Mewtwo. Even then, it was supposed to be uncatchable and it really serves little other purpose, in most media kinda just showing up and existing in Kanto. Jirachi is sleeping and serves to just be an interesting Pokemon in the game. Manaphy is a reward for a spinoff, Arceus is a God but also doesn't really have a defined part of this world.
Shit, even after an entire Arceus game, its role in the world is not really shown to be clear at all. In fact they basically confirm that the Arceus we know isn't even its full form, and we will never know what it is. My problem with most Mythicals post Gen 5 isn't that they're not placed into a point in the world, since in my opinion the best mythicals are the pixies with incredible power that don't actually do much.
Mew, Celebi, Jirachi and Victini. The DNA of every Pokemon, time travel, granting wishes and guaranteed victory. These are all insane powers, more powerful than Darkrai by a mile, but they didn't have them just maliciously doing things to humans since that doesn't make sense for the Vibe. These people aren't normal Pokemon, ie. they aren't entrenched into the world's regular ecosystem, and they're not legendaries - they aren't a part of the massive collective of Pokemon that impact the world day to day.
I think after Gen 5 mythicals as a status kinda started sucking and I still think mythicals in gen 5 mostly sucked. Genesect is literally just Gen 5 Mewtwo, why is it now a Mythical? It's the same story basically. Would Mewtwo be a mythical? Volcanion doesn't seem to be very elusive frankly and it seems to do less than Heatran.
Hoopa is I think one of the only modern mythicals that really fits the bill of what the term used to mean. I like Magearna as a Pokemon, but it being a mythical doesn't seem to make sense to me, and how you get it is boring and lame.
I want mythicals and humans to not really interact much outside of being caught by the protagonist if that's even what it's about. If you think about the Pokemon world's ecosystem as different audio channels, mythicals should be invisible and not interact on either the channel with legendaries or the normal ecosystem- they should be their own weird oddities, reminders that this world can have a time traveling God pixie right behind that tree and you wouldn't even know it.