Hmm...while I'm perfectly content with a spell slot psion, I wonder if approaching this from another direction would be useful.
Spell slots are one way to do power scaling in 5e. The other existing ways are Extra Attack (sometimes in combo with Spellcasting) and Sneak Attack. If we focus on the system and yeet everything that uses spellcasting and see what fiction emerges we might get...
- Psionics As Attacks. In this model, we maybe focus on attack/defense modes as weapons. Ego whip, mind blast, etc., all get presented as things you can use with the Attack action to hit with. Power scaling comes by giving, basically, Extra Attack and Action Surge like a Fighter. Each of the attack modes is like a weapon - deals damage and has a mastery/cantrip effect on top.
- Psionics as Sneak Attack. In this model, we just deal extra dice of damage. It's possible the subclass defines the context and type of damage (ie: if I'm a telepath, maybe I deal extra psychic damage when I hit; if I'm a telekinetic, maybe I deal extra force damage and also do some forced movement/knock prone; if I'm a seer, maybe I deal extra psychic damage from a destiny dissonance, but it's more like 1/encounter than every round).
- Psionics as Rage. In this model, we rip from the Barbarian - we get juice from Extra Attack and from entering a psionic state that helps us deal more damage with our attacks. Our subclass also adds some extra damage. It's a bit more of hodge-podge of abilities, but the basic thing is that in the end, I'm dealing damage thanks to hitting with attacks while buffed in some way. The "rage" is really an opening of the mind, a third eye awakening, a shounen revelation, whatever. This is kind of a cool idea for a wilder.
- Psionics as Flurry of Blows. This works a lot like Psionics As Attacks, but the focus is on a bonus action to make many smaller attacks. You can still use the attack modes as "weapons." You still get Power Points, they're just roughly equivalent with a monk's Focus Points. Subclasses might focus on expending Power Points to do dope stuff.
In all cases, subclasses focus more on "utility" -- your telekinesis and your telepathy and your teleportation and your precognition and such are mostly defensive / social / exploration use. These can be balanced for at-will or nearly at-will (short rest recharge) capacity, since they're not exactly sources of the class's damage potential.
Of the potentials above, I am fond of the idea of treating psionic attack/defense modes as weapons, and so Psionics As Attacks or Psionics as Flurry of Blows are pretty attractive. I also like bringing the monk's design into the psion, since that reinforces the occasional idea of the monk being kindred, but not quite the same, as a psion. I think it could be a lot of fun using the monk as a template to make a psionicist.
Ultimately, the spell slots are probably the most evocative of the 2e/3e/4e psionicists, but I totally think there's room for "psionic weapons" (with defense modes being "psionic armor!") used by those proficient in them, and lots of room for psionic subclasses of different classes. The former could even give a very 1e "I'm a Fighter, but I can also Mind Blast!" kind of vibe.