Who is Alphinaud and who is Gaius?

[Cilia]

As for why primals are such a big deal, and why the empire has been spurred to fight them..that’s a later detail!

This is actually explained very early in the story.

Primals are a big deal because they are false deities that drain the world of aether (energy) just by existing on the material plane, and further because they have a tendency to temper (brainwash mind-slave) people in order to give them an anchor on the material plane, thus ensuring they will still exist even in the aetherial sea. (The desire to temper is considered a core feature of primals when first introduced, but numerous primals since introduced have shown an aversion to or apathy toward tempering.)

The Garleans were spurred into opposing them because the first Emperor, Solus zos Galvus, saw the devastation that results from repeated summons / prolonged primal presence in the form of the Burn, a large swathe of land completely drained of aether, which reduced it to a desiccated wasteland.

Hide boxes are spoilers - don't click on them if you don't want to be spoiled.

The reality is that while most of the above is indeed true, the Burn was not created by primals - that was one of the key lies spun by Emet-Selch (a.k.a. Solus zos Galvus) to start the Garlean war machine in perpetual motion. The Burn was actually created by an ancient Allagan device (located on the Azim Steppe, as "The House of the Crooked Coin") that drained the aether out of the land so it could be excavated and launched into the sky, becoming what we know as Azys Lla (visited at the climax of Heavensward and sporadically thereafter). The device was left on because the Third Umbral Calamity all but annihilated the Allagan civilization. Emet-Selch is aware of the truth, however, because he was the one who set up Allag as well.

That said the Garlean method of dealing with primals, like many things about the Garlean Empire, is designed to be counterproductive. Not only does the military conquest move people to summon out of desperation, the enforced atheism and brutal rule further engenders summoning in bids for liberation.

While it's said early on that all primals desire to temper people and stay on the material plane to amass as much power as possible, numerous primals since introduced do not display this behavior. Ramuh tempers only those in his immediate vicinity upon summoning (as he can't control that) and those sylphs who ask him to do so, and gracefully accepts his defeat knowing he is a danger to the material plane. Bismarck appears too bestial to temper people. Not only does Alexander never temper people, he engineers his own defeat in a(n almost literal) game of 4D chess and then seals himself outside of spacetime. Susano'o is more interested in getting a good fight than tempering people or being worshiped, and also gracefully accepts his defeat. And Shinryu is too consumed by a lust for violence to bother with things like worshipers; he actually does try to temper people, once, but only gets the PC and Zenos in his crosshairs, both of whom are immune to tempering anyway.