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If this guy is anything to go by, it’s snark.
Except that Alfred Bester isn't particularly
charismatic at all. He's refined, certainly, but he's actively and intentionally abrasive to most of the people he meets, with an air of smug superiority rather than being compelling. In most cases where he gets what he wants, it's because he backed someone into a corner (e.g. Lyta being allowed to pretend to be a Corps telepath), or because he's already got friends, or more commonly blackmail, in high places (e.g. Psy Corps is in bed with Night Watch and President Clark), or because he literally had someone's personality modified to be his perfect traitor puppet.
Instead, Bester is a chessmaster. He plans three or four steps ahead. He doesn't compel people; he
coerces them. If I were expressing this in 5.5e terms, he has expertise in Intimidate and perhaps an Origin Feat which includes allowing him to substitute Intelligence in place of his Charisma modifier for Intimidate checks. It is his ruthless, devious intelligence that makes him both feared and dangerous (well, that and being a P12; telepaths from P8-P9 and up can implant thoughts or directly induce behavior in "mundanes", so they don't really need to be
charismatic to make people do what they want them to do.)
Likewise, there's another character whose telepathic power was transformed via experimentation, Matthew Stoner. He ceased to have most typical telepathic abilities, and instead developed "empath" abilities, allowing him to mold and shape others' minds subtly, rather than the "brute force" manipulations of strong telepaths. Unlike Bester, Stoner
is shown to be at least somewhat charismatic, but his charisma still seems secondary to cleverness and subtlety when it comes to manipulating people.
Also, another example of psychic powers, albeit under other names?
Dune. Melange specifically grants
prescience, and I'm pretty sure both the books and the films make reference to either "psychic" or "psionic" abilities at some point. Certainly, "the stuff Bene Gesserit and Paul Atreides do" is not going to be a difficult sell for current-day teens and twenty-somethings.
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Now, all that said? I could see psionic powers perhaps working differently from spells in part by not having a single consistent base stat, but rather having different stats for different purposes. E.g., let's say we have the following psionic "disciplines", each with its own core stat:
Kinesis (Int)
Mesmerism (Cha)
Cryptaesthesia (Wis)
Psychomorphosis (Con)
Different abilities would fall under different disciplines. Thus, the Psion does not really have a "core stat". You have to specialize, or select powers that don't strongly depend on the relevant stat. Kinesis is physical offense and manipulation, e.g. pyrokinesis, levitation, technopathy if relevant, Bloodbending-like body manipulation, etc. Mesmerism is psychic powers used to beguile, confuse, or control minds, thoughts, or perceptions in other beings (or yourself) and to defend yourself against such things. Cryptaesthesia is clairvoyance (and all the related things), remote viewing, prescience, "mind palace" stuff, etc. Psychomorphosis is the use of psychic powers to modify or manipulate bodily processes, the link between mind/soul and body, and giving inanimate things "life" in some form, but might also include teleportation (it could go here or Kinesis, sorta depends on how it's interpreted).
This would recognize the important place Intelligence has always had for psionics, while still recognizing your point that a strict relationship between mere IQ (already a dodgy thing on its own) and "mental muscle" is fraught and probably outdated. The "Kineticist" type of psionicist would focus on Kinesis and Psychomorphosis, while the turn of the century "Medium"/"Spiritualist"/etc. would focus on Mesmerism and Cryptaesthesia.
From there, you could design subclasses that emphasize specific stats and/or disciplines, in order to evoke a particular era's view of the paranormal/psychic/etc., or to give representation to a particular work's take on psionics, etc. A "Mystic" that focuses on Cryptaesthesia and Psychomorphosis, being akin to the Oracle at Delphi? A "Hypnotist" focusing on control and (mental) support effects in Cryptaesthesia and Mesmerism? A "Paranormal Investigator", perhaps, that focuses on the application of psionics to skills? I'm sure we could come up with more concepts, and that's without digging into past editions to see if we find any nice stuff.