This strikes me as a false dichotomy. This is not really an either/or scenario. For example, in your latter example of "psionics are different," psionics could use a spell system but not have dispel magic or other regular spells apply. But in the former example of "psionics-is-magic," dispel magic and such could still apply but with the hypothetical psionicist having different mechanics than the regular spell system.
It /could/, hypothetically, sure. But, 5e tends to give different things different mechanics, even when the differences may seem like something that's arbitrary and could be done with just re-fluffling.
And, re-fluffing spells as 'psionics,' even when accompanied by a ruling that they're, in essence, spells that are immune to dispelling ('broken' though that arguably might be), still has mechanical implications. Components, for instance.
That said, I don't think that your suggestions of just reskinning or creating a new subclass of preexisting class (e.g., warlock, sorcerer, wizard) would work for a psion/mystic.
Not my suggestion, rather, I was holding it out as something that /wouldn't/ work for the psionics-is-different feel, at all. Since psionics has been different or optionally different in every edition, it's not a viable solution, on it's own.
At least no more than reskinning a fighter, bard, or cleric to make a warlord.
Exactly right, and for the same reason. Re-skinning spell-casting class to act as a non-magical class is simply fraught. Even if they're accomplishing some of the same things, they should be doing so quite differently, if they're to fit in the broader 5e design aesthetic.
Perhaps psionics should be considered "different" in much the same way that arcane and divine magic are considered "different" from each other within the traditions of D&D? (I also do think that 4E probably had the right approach by grouping the monk and psionics together.)
I was not at all sanguine about dropping Ki as a power source at the time - even though, personally, I never cared for either the monk nor psionics, it seemed like a cheap trick to combine them like that. In retrospect, I suppose, it was OK, and it did lead to conversations that spelled out what was ethically wrong with D&D's traditional "orientalist" treatment of the Monk, so that was enlightening.
In the case of 5e, though, Ki is firmly established as actually being magic of its own stripe, so Ki-as-psionics would imply psionics-is-magic, which is really something I think should be left in the DM's court, as 5e is the DM-Empowerment edition.
If it breaks your fiction, then maybe you had the fiction of the fantasy world you envisioned wrong to begin with. So perhaps it would be worth your time to reevaluate your own sense of the fiction as a player rather than accusing the class, the other player, or the world of being wrong or so insensitive as to break "your immersion." It's basic table etiquette. It's the GM's call and not the players'.
It's not 'wrong,' no matter which way you slice it, whether you're the odd player out or coping with an odd player out or the DM dictating how it'll be at your table. It's just something you have to reconcile one way or another. I feel like there's no need to make a big deal out of it when you're the odd player out, just keep your opinions and visualizations in your head and let everyone else enjoy the game.
having to edit personal fluff at the table is nothing new at all. Nothing new, so acting like having to do this for psionics requires extra effort.
There's a lot of things at the table that are going to be imagined a little differently by each player present, regardless, without anyone even noticing. It's a shared experience in concept, but it can't be a perfectly consistent one.