I think the issue with Psionics is, you can't design it for people who "don't want Psionics".
The issue is that a lot of people don't want just a half-caster with a sword (available already as an Artificer Battle Smith), they want something like the Swordmage of 4E, where the magic is fully integrated into their combat, where they're not just sometimes casting spells. Or at least the Magus of PF2 (not PF1).
Bladesinging fits well for an OD&D/AD&D-style Gish i.e. "Fighter/Mage", but less well for what a lot of people want.
See, in 4e, the traditional half-casters were fully integrated mesh between martial and magical (see Paladin, Seeker, Artificer, Bard, Assassin) or else made a choice and were completely unintegrated (Ranger). Later on in Essentials, they backtracked that and did mixed-power source classes (Cavalier, Scout, Hunter, Blackguard, Skald, Berserker, etc), and 5e returns to half-caster progressions like 3e had (though not for Bard, who got to stay fully magical).
A lot of people when they say they want a fully integrated class in 5e, what they mean is that they want an Eldritch Knight with Bladesinger spell progression, which is broken for the balance of 5e. It worked in 2008 4e because everyone had comparable AEDU power progression. But in 5e, extra attack and fighting styles and martial weapon and heavy armour profs are balanced against higher level spell access (or at least are supposed to be). The system is built toward Gishes being half-casters, or else being full casters that aren't nearly as martial as people are asking for (Hexblade, Valour Bard, Swords Bard, Sorcadin, War/Life/Nature/Tempest/Trickery/Death/Order Cleric, etc).
The only class that truly is comparable to Paladin and Ranger here is Artificer, and Swordmage fans are frustrated because (1) the very clearly swordmage-esque subclass is locked behind setting-creator hellcow's best-selling homebrew Exploring Eberron
and not technically canon; (2) the "canon" warrior Artificer subclasses are Sword and Board and Doggo, or else Iron Man (neither of which is teleporting magic aegis swordmage); (3) the base class isn't designed as a martialist by default and people don't realise the swordmage is hidden inside of the Artificer; (4) Artificers get cantrips by default instead of requiring a fighting style choice to access them.
The Artificer very clearly has a class narrative that defines the gish in a way that doesn't step on the toes of Bladesinger, War Mage, or Eldritch Knight, and yet still is an Arcane Warrior (Magitech Warrior, specifically), but I'm willing to bet that if the Battlesmith and Forge Adept and Armourer were pulled out and made their own class, with cantrips as an opt-in Arcane Warrior fighting style instead of by default, people would be less bothered by the "lack of a true gish class".