Notice the gishes:
Arcane gish: Alchemist.
Divine gish: Paladin.
Psionic gish: Barbarian?
Primal gish: Ranger.
The Barbarian is magical, but not a spellcaster. Probably, by definition, a gish must be a part-caster with spells. In any case, the following explores if the Barbarian or something like it would function as a "psionic gish".
2024 flavors Barbarian with "primal", but it can be understood as psionic. In the Norse context, the berserkar are psionic. They do go out into the wildereness to commune with animals in nature, but it is in order to become an animal mind − and body. They are shapeshifters, typically a specific animal. The boundaries between humans and other animals are blurry and porous.
Barbarian Rage is a mental effect, and the use of Constitution for AC connotes metamorphosis. It can easily be psionic magic.
In any case, psionic and primal are two sides of the same coin. Psionic is the power of ones own soul. Primal is the power of the souls of the nature beings across the Material Plane. In this primal sense, the mountain is a soul, and so on. Again, animals are blurry. The main difference between psionic and primal is thematic emphasis. Psionic is personal, individual, and feels human. Primal is relational, communal, and feels elemental. In this context, plants are aspects of elemental souls, like the branches of Yggdrasill being the domeshape of the sky, and forests growing from the earth like hair. For D&D, it helps organizationally to treat plantlife as if a living element made out of the other elements.
For the 2024 Barbarian, the Berserker subclass is also psionic. The magic is subtle, and doesnt actually shapeshift. It would be nice to somehow have an alternate Beast form and travel ethereally. To be fair, there are also Norse stories about these subtle kinds Berserkr warriors, in addition to the ones that are overtly magical.
The World Tree Barbarian subclass is definitely primal, despite its reference to the Astral Plane and its "Outer Planes". Properly, Norse Yggdrasill is the concept that the universe (the Material Plane) is a single living organism. It flourishes and withers with the cycle of seasons, and interconnects everything to everything. It has three roots, which shamanic trances utilize for outofbody journeys. One root goes directly north to the North Pole, and from there downward to the underground. Another root goes south and east to the location of the sunrise, then up into the sky, and another goes south and west to the location of the sunset identified with the þursar, dangerous but magically knowledgeable land nature beings. The branches are sky. Apparently, the location of the trunk of this tree is subjectively near any grave and at the center of the round horizon. The concept of the world tree is a blend of Sámi polar-axis shamanic traditions and Norse tree-burial traditions. Yggdrasill is the Material Plane, and perhaps there is a literal tree visible from the Ethereal overlap. That said, it might plausible to say "some of the tips of the branches and roots reach the Astral Plane". To disassociate the mind-soul from the body-soul, one shamanic technique is to visualize the death of the body, and the body as a corpse. The hypnotic mental perspective can then adopt an alternative concept of self, and elsewhere. Via this shamanic meditative self-visualization, the being of the sky "sacrificed himself to himself" by hanging himself on the worldtree. This can be characterized as self-transcending mode of cognition. While free from the body and the world, he intuited the nature of the written language of the runic alphabet. This linguistics that underpins and constructs the experience of the reality, feels "Astral" in the D&D sense, and the world tree is a journeyway to this experience. Nevertheless, the World Tree Barbarian subclass has teleportation features, without planeshifting, thus feels appropriately Material, animistic, shamanic, and primal.
The Wild Heart subclass self-identifies as a member of a family "kin" of animals. The text expicitly flavors "primal powers", which is fine, since it also emphasizes the community of animal souls. A more psionic emphasis might be the personal metamorphosis into an individual animal, mentally and physically. A shapeshifter.
The Zealot subclass is clearly the divine power source, referring to theism and "priests", and the "ecstatic" experience of Astral influences. Even if the Barbarian is a psionic gish, it is ok to have subclasses belonging to other power sources. The martial Fighter has subclasses that include arcane and psionic, for example.
If the Barbarian is to be psionic, other subclasses might emphasize this. One of the subclasses might a spellcaster. But also, a Barbarian subclass seems a helpful location for a "shapeshifter class". The concept of shapeshifting is simple enough, perhaps even a single spell. But a subclass can more fully express it, and scale it well into higher levels. This shapeshifter Barbarian might emphasize the flavor of psionic metamorphosis. Perhaps there is a favored alternate shape, but other shapes are also possible, and medial transitional shapes in between.
For the arcane gish, the Alchemist. I like the alchemical flavor, it is spot on for "arcane". I would like to see it more like the Ranger, where some subclasses have a "pet" but the class itself doesnt. I would also like to see a more Paladin-esque arcane warrior, like Eldritch Knight but more overtly magical, and moving swiftly in alchemical heavy armor.