To be honest, this seems like a punt. That may not be a bad idea.
Psionics is now a descriptive term, like Arcane and Divine. We still think of certain classes as "arcane" (wizard, bard, sorcerer, warlock) or "divine" (cleric, druid, paladin, ranger), but really, those terms are holdovers from earlier editions and fluff; both terms are mechanically meaningless. Further, classes are mutable: a sorcerer or warlock might normally be considered arcane casters, but the divine soul and pact of the celestial make them divine powered with no change to their mechanics. Lastly, plenty of abilities could be considered Arcane or Divine without actually being spells; a cleric'c channel divinity or a Eldritch Knight's weapon bind.
Psionics is the same: it describes the power source of certain casters and effects without changing the mechanics. An aberrant soul sorcerer is the psionic equivalent of the divine soul; a change in power source with some new subclass features and spell options but ultimately mechanically the same. Certain powers and abilities (like a mind flayers psychic blast or a kalshatar's telepathy) are likewise "psionic flavored" but the descriptor has no mechanical bearing.
Perhaps down the road, a full psionic spellcaster class will come to exist (Dark Sun?) But I think it will be a spellcaster and not a new mechanic. However, until then think of psionics as fluff, not rules much like how arcane and divine are fluff and not rules.