I really don't think we should keep the tradition of Psionics doing explicitly spell-like things. Storm magic just is not Psionics, I'm sorry. Psionics may sometimes involve storm aesthetics, but explicit storm magic, or naughty word like transport via plants, these really make psionics an Aberrant Mind with almost no difference in fluff or mechanics.
If you look at the 3e Psion class, you can see that the Psion really is spells.
That said. There are things that the Psion class needs to be able to do that 5e does not yet have a spell for.
Also, not every 5e spell has a spell description that feels psionic in flavor. A spell can feel nonpsionic if it is too complex and fiddly (Prismatic Spray). A psionic spell effect needs to be a salient somewhat simple and coherent concept that it easy to visualize as a whole. Similarly, a spell that explicitly relies on the power of some other creature to do the effect (Planar Ally) or gets too distracted with ritually handling a object (Magic Jar) feels too external to be ones own mind doing it.
The psionic spell lists also avoid themes of undead and necrotic. Despite the fact that a ghost is a kind of disembodied powerful psionic mind, the deathly negativity is like veil distinguishing it from a living mind.
I probably agree with you about Transport via Plants feeling less psionically flavorful, in the sense that it is somewhat complex and fiddly. But I included it in the first pass to see how many 5e earth-plant spells there are.
Relatedly all elemental spells are in the same category. If one feels comfortable with a "pyrokineticist" "fire starter" character, then an other player will feel comfortable with an "aerokineticist" "X-man Storm" character, or a "geokineticist". Indeed, 3e Pathfinder has all of these element-manipulating psionicists, as class options.
Personally, I prefer all elemental magic − fire, air, water, earth, and plant − to split away from the "psionic" power source and instead be part of the "primal" power source. Psionic and primal relate to each other and are even both mindful powers. But the elemental themes of primal can be kept distinct enough from psionic.
All said. The 3e Psion is a spellcaster choosing known spells. But each discipline needs a curated spell list that feels psionic and behaves in ways one expects a psionic effect to behave.