I've put most of this in other "Psionics" thread...but since it's relevant I'mma post it agin.
First of all, I think some (if not most) of the issue folks have with Psionics comes from the name. "Psionics" is born from the idea of "Psychic-Bionics"...enhancing the brain to have these extra/psychic powers...using more than 10% of the human brain and all of that. It is a decidedly old school sci-fi pulp trope when the lines between sci-fi and fantasy were decidedly more blurred as genres than they are today. Giving the space-characters/alien races something that was magic-like with a "sci-fi-tific" explanation. Telepathy, Telekinesis, Clairvoyance are things that can all be and feel "real" is a sci-fi universe...but Magic Spells and Sorcery? That's just stuff and nonsense that people who didn't understand or know any better came up with to explain psychic phenomena. The "paranormal" is fine and dandy...the "supernatural" is bunk. Think the X-files vs. Supernatural...or Buffy the Vampire Slayer, for that matter. Star Wars vs. Star Trek (or the current Marvel movies' explanations). The repeated trope that what appears to be "magic" is just technology we don't have/haven't figured out yet.
Changing the word/name in D&D, specifically, to simply "Psychic" and referring to "Psychic/Mental Powers" would do a LOT, I think, to mitigate a lot of the misunderstanding and displeasure folks have with "Psionics."
Beyond that, I agree, for space and simplicity, that anything Psychic powers can do that have a magic spell equivalent, should just be the spells. I don't need "Charm Person" and "Sleep" and [something like] "Personality Influence" and "Remove Consciouness" as separate entries that do the same thing. The Psychic simply has a different, some might say "personal/inner magic" or "enhanced mental awareness" or "born this way" or "battery/receptacle of the collective psychic energies of conscious beings -which most don't know they have, let alone can control or use"...however you want to fluff it...they have a "casting" (another word choice causing confusion issue) method that mirrors some other spellcaster, but not exactly/isn't the same thing.
Sorcerer's have spell points? Give the Psychic class Psychic Points. They don't need 'slots" other than how many Points a particular power manifestation would cost. No "tiered levels" of what they have access to. If they have the points to spend, they can make an effect happen. You're using a system that's already in place. I don't need an entirely different mechanic...just not an exact copy of the sorcerer system...no slots...no "prepared spells"...
And they should have, probably as a class feature/ability, the power to push themselves beyond their power points...giving themselves physical and/or mental harm in exchange for going that extra mile when the emergencies require..burning Constitution or Wisdom (or simultaneously?) or even HP makes sense to me. With built in recovery rules, obviously. But the distinct possibility that you could 'burn out" your powers for a period of time...if not actually kill yourself by going too far.