As I've already said, you don't NEED a new set of rules to play a narrative Psion. You can already do this by reskinning an existing class. There is already a subclass of sorcerer explicitly written to fill this niche.
While you
can, it is not the best approach. We could just have 4 classes, too.
There already isn't much daylight between sorcerer and wizards as designed.
They have some structural similarities, but I find them to play mechanically differently due to the metamagic. Further, the RPG hooks of each class are VERY differernt. If you find that playing them is very similar to you, then there is a large part of the D&D experience that is available to you that you're not capitalizing upon.
I'm not sure how you could wedge a third full caster using spells and slots without crowding into an area that doesn't have a lot of space available to make things feel different.
I've addressed this numerous times throughout this thread. I've been doing it for 30 years.
In my setting, a wizard uses their intellect and mental prowess to pull magic from the spell weave, craft and shape it, and turn it into their spells. A sorcerer's [heritage/taint] allows them to pull magic from the weave and through force of personality, force it into their spell. A psion generates power within themelf and then crafts it into their psionic abilities.
Psionics do not tie to the weave, and thus do not interact with magic that is built on the weave connection such as anti-magic (such as a beholder eye), detect as magical under detect magic, or get beaten by a counterspell. Their abilities, historically, have not relied upon 'spells' but have instead been crafted differently. The psion has a 'metagame' aspect when they battle other psionic creatures with the various psionic attacks and defenses being a 'rock, paper, scissors' (or boulder, parchment, shears) game where you can invest in a wider variety fo defenses at the cost of limiting your offensive capability, or you can use fewer defenses and gamble that the enemy won't attack with the right psionic attack to pierce your defenses more easily.
Psions, historically in my setting, do not play like wizards. They play like super heroes/villains, like Professor X, Jean Gray, Martian Manhunter, or the Shadow King. When a psion faces a non-psionic character, the distinction is lessened, although as I do not generally use 'spells' as a template for their abilities they are distinct. However, when two psionic creatures face each other in battle, it is a distinctly unique situation.
They're also, due to the lore of my setting, tied back to the Far Realm (as it is now known - this is the modern evolution of the lore in my setting - it used to be a bit different but that story would take a lot of explaining), which gives them a distinct connection to many aberrations and a connection to the Cthulhu mythos, which is a
MAJOR player in my setting's big storylines.
In 5E, I have not fully implemented psionic rules for players to use and have not encouraged psionic PCs as I have WAITED for official rules to be provided so that I can adapt from them (as using mainstream psionics would be more approachable than teaching people my unique system by itself), but it is there behind the scenes in my psionic monster designs, and in the psionic NPCs that the PCs encounter.
Regardless, if you think there is no unique design space around which to have psionics be a meaningful, distinct, engaging and beneficial element in your D&D games, many of us have decades of evidence that proves that notion wrong.
