The first thing that it is important to remember is that there is (game-mechanically) a false limitation on the spells you know. The names of magicks attached to certain effects and the repertoires held by specific disciplines change greatly over time – and yours are limited for game balance. There’s no reason a Black Mage can’t know Stoneskin, for instance. Y’shtola would be thought of in today’s terms as a conjurer, yet she knows several spells outside of its purview, even one that’s taught specifically to astrologians – because she’s from Sharlayan.
The second important thing is that the spells you’re mentioning aren’t summoner-based, they’re arcanist-based. Remember how few actually come from the job menu? These are the spells taught to present-day Lominsan arcanists based on what manuals survive from past users of arcanima-like magicks (especially from Nym and the isles of the south) and the zeitgeist of contemporary research.
As an example of how deep-seated this is, realize that egi behave identically to Carbuncle. Only the names and animations are different. Garuda-egi behaves almost identically to the wind-based Emerald Carbuncle because, while you may have resurrected the arts of Allag, its still heavily rooted in what you already know. You’re basically re-inventing some of the sacred geometry usually used to give form to the latent power of gems to instead conjure forth the taint of Garuda upon your own aether.
In Limsa Lominsa, as it was in Nym, the aim of the game is strategy – have a practiced plan ready to engage any situation. Every arcanist should have a playbook – a lexicon of all geometrical patterns used in aforementioned plans (an… omni-lex, if you will). You’ll even see Lominsan arcanists sometimes combining the strategy and math to calculate the odds of success in employing a specific stratagem.
To gain an upper hand in combat, the arcanist (armed with a book and requiring time to pen out their geometry) needs to enfeeble stronger, more agile opponents coming at them with physical weapons. A good way to do that is to make them ill – to conjure forth maladies that slow them down, blunt their healing, sap their strength, and leave them wasting away.
As a modern-day summoner, you just come with a background in contemporary Lominsan arcanima. The job skills merely teach you how to put more oomph in it and how to infuse your aetherial constructs with primal essences (until you get to 50, where you start learning some of the specifically Allagan themes, which you focus on the ultra-powerful, recently-deceased Bahamut – whose tainted aether is already just… everywhere). However, you’re still a fusion of past and present.