He's not a Psion because he can't heal? That's a long bow to draw.
I just honestly don't even slightly for one heartbeat agree. I've been healing people with Psionicists and Psions since 1991 dude. This is not negotiable for me. Healing is part of Psionics. It's also part of it in fiction, very consistently, even in Star Wars, as we recently saw!
As well as Biofeedback, the other big thing people bizarrely ignore, but which is common in fiction and in previous versions of the game is time/space stuff - i.e. teleportation etc. - but Wizards have good access there.
And he's not memorizing spells in the morning. He's meditating on his Psionic Crystal and attaining Psionic focus.
You don't seem to understand the issue. The issue isn't how you get them in there. It's that you have to operate like a Wizard, i.e. having only specific spells "armed" each day. It's not the Psionicist/Psion model at all. Not in previous rules, and not in fiction. Psionicists/Psions have consistently had a more limited arsenal, but access to all of it all the time - and there's loads of classes in 5E that DO work that way. Just not Wizard.
I'm actually kind of interested here, not trying to be mean, like this is a real problem. Wizards are not the right base. We could work out how to build a 5E Psion, but we can't do it from Wizard.
But I otherwise agree. I could fluff a Bard as a Psion also. I could fluff a Paladin or Ranger as a Psionic Warrior as well.
Refluffed Hexblades make for pretty compelling Githyanki 'Antipaladins' also - the patrons name is changed to the Lich Queen, and the Sword they bond to for + Cha is a silver sword.
Yes that's a much better refluff than actually using the Fallen Paladin or Conqueror Paladin or the like, I agree (no sarcasm, I actually do agree).
Paladin would mechanically be an awesome starting point for 5E Psychic Warrior, if you just like, overwrote the fluff entirely. Fighter is a dodgy one. It's not a disaster but it's not where it needs to be.