I've used Psionics in my 3.5 game, where they were finally OK (2e tried to fix them, but implementation was off even for DarkSun). I've also played a Fighter/Mystic recently in 5e (v2, then v3), and it had some big problems still IMO including what the top 10 or so levels should really be about.
For me, Psionics work best in D&D when they are:
* a core part of the game i.e. an option for players, NPC, monsters etc
* different in flavour and implementation, but only slightly different
* balanced vs spells - especially allowing psionic and spell effects to be 'the same' i.e. dispel magic works vs psionics etc
i.e. an option that's essentially "spells using the mind", and it's a Class or Monster Feature (e.g. Psionic Mind Flayer, Dragons might use Psionics instead of Spells, etc), not an add-on.
For 3.5, the saying that stuck with me for Psionics was: "this is how a Sorcerer should be", i.e. the Psionics Point pool, which can be used to manifest(cast) whatever power(spell) you had access to; and the ability to Augment a power.
In 5e, we already have Augmentation for spells. DMG has a Spell Points system, and not surprisingly that's what the Mystic uses. Where the Mystic fails so far, is that it is not designed clearly in terms of spells vs psionics, and the 'differentiation' between a Psion vs Spell caster is mainly that a Mystic ends up being "jack of all trades, master of none" - as they head to 20th level, they learn most powers, but none of them are spectacular. If we could tighten up the flavour differences between "core psion vs core spell slinger", and give more focus and power (not breadth) to the Mystic as they level up, and make it compatible with multiclassing (some seriously OP dip potential as written, and spells vs psionic points if you have access to both); well, if you could fix all that, then I'd love to see Psionics back in 5e.
As far as game Settings go, while Psionics were a core part of Dark Sun, I feel they could easily be part of any other game world, it's just that traditionally they have been a bolt-on after the fact, instead of embraced and integrated early on. Have fun trying to work it into the Forgotten Realms though - maybe they arrive on the Sword Coast soon via Chult?