I'm not sure I agree with this assessment of the disciplines. In brief:
Conjury is communion with nature and the elements - and to a lesser extent, the Elementals. Most Conjurers never directly witness an Elemental, but all Conjurers meditate on the elements. To a Conjurer, each element is important, and each carries a specific meaning. A Conjurer's meditations are contemplative and philosophical, feeling the element-tinted world around them and from which they draw their power.
Thaumaturgy is mental discipline. Despite its its element-aligned spells, it has nothing to do with the elements philosophically. To a Thaumaturge, Fire is a tool for burning things - nothing more and nothing less. Because their power comes from within, Thaumaturges are very concerned with how to increase their potential. a Thaumaturge's meditations are forceful and introspective, controlling the ebb and flow of aether within their own soul.
Arcanima is precise application. Arcanists study and refine how to shape and apply small amounts of aether for maximal effect. Grimoires are toolboxes filled with ways to more finely shape aether than they could with their minds alone. Arcanists don't meditate, because aetheric capacity isn't much of a concern for them. Instead, they spend their contemplative time practicing formulae, and staying up to date on the latest techniques and literature.
Arcanima -> Thaumaturgy/Conjury: Arcanima can't create fire out of the geometries, because the amount of aether that goes into a fireball would overload their books. Fireballs are sledgehammer compared to the razors that Arcanists are used to swinging.
Thaumaturgy/Conjury -> Arcanima: Thaumaturgy and Conjury can't be used to summon Carbuncles, because their tools (mental discipline or the blessing of nature) are too clumsy for the fine-detailed manipulation needed to create a Carbuncle.
Thaumaturgy -> Conjury: Techinically possible, but hazardous to the caster. Ask Sylphie's mother how this one worked out for her.
Conjury -> Thaumaturgy: I tend to RP that Conjury can use all 6 elements (but doing so is not the same for them as it is for a Thaumaturge - a Conjurer can't force an Astral shift in Nature to increase the force their spells, the way a Thaumaturge can in their own soul). Conjurers tend to be most practiced in the ones that are necessary for day-to-day maintenance of the Shroud, hence Earth, Water, and Wind. Conjurers don't practice with fire or lightning because no one likes forest fires.
All this doesn't mean that a Thaumaturge can't cast Virus. They clearly can (game mechanics are lore, too) - but they're using Arcanima when they do it (with the geometries inscribed on a glove or a plate on their staff), not Thaumaturgy.
Also, none of this applies to jobs. WHM seems like it would have the same limitations and CNJ, and SCH seems like it would have the same limitations as ACN, but SMNs absolutely can use elemental magic, via Egis. BLMs theoretically could draw on the planet's life force to fuel Cure spells without burning themselves out. I don't know enough about AST to talk about them.