I think we may be working off a different definition of 'psychic'. For instance I see the Dark Eldar and their soul eating as no different from a more refined version of Psychic Vampirism, which is the only thing they can use without fear of She Who Thirsts, potentially because it is the one power that makes them more like Slaanesh, they are sort of making a sacrifice and acknowledgement to the God that is most of their pantheon... because it ate most of their pantheon.
The thing is; there's nothing that I can find that even implies that may be the case, and a lot that says that most Dark Eldar use a combination of drugs and rigorous self-control to suppress their psyker abilities, but nothing that suggests that does anything to influence their ability to eat emotions or souls or super-charge themselves with the consumed energies, which is just a different ability. Similarly, Craftworld Eldar who are on a path that suppresses their ability to use psyker powers so they can't draw on the warp have an inherent talent to regenerate their wounds that can be triggered by healers who can induce the right state in them, even though they can non longer trigger it themselves.
To use a WFB example, the Dark Eldar's powers seem a lot more like vampiric Blood Gifts, which are also usual fuelled with stolen energy, than they are like spell casters calling on an external source of magic, who are more like psykers.
And to use an in quest example. An enlightened mortal can choose to use their own Essence to fuel a psyker ability, rather than using their Willpower to draw on the energy of the Warp at the risk of Perils.
Similarly, Dark Eldar and Navigators can use much more limited psychic abilities that they power with their own souls without needing to use a psyker's ability to tap into the Warp for power.
Basically, they have something like inherent Gifts or mutations that either don't require external energy, like Navigators' powers, or can themselves be used to steal others' energy, like Dark Eldar.
Tl;dr: Psyker powers are defined by the energy source. Psykers draw energy from the Warp. Neither Navigators nor Dark Eldar do, albeit for different reasons.
The Navigators literally have a third eye that sees into the warp, how in Khornate cotted tomato juice that's not a psychic ability?
Psychic is not the same as psyker.
A psyker's soul is a gateway to the warp through which energy spills, energy they can direct into psyker powers. That's why daemons can more easily possesses them and Enslavers can widen the gateway to physically enter the Materium through a Psyker.
A Navigator's soul does not contain that gateway. They can perceive and influence the Warp, but they aren't an unlocked door that they need to hold shut through Willpower or a patron's blessing like a psyker does.
