We don't know why the Allagans sealed Bahamut away and sent him into orbit, but the easiest answer for this is: The furthest he's away from the Eorzea, the less likely he can absorb aether and become stronger. I suspect the Lunar Transmitter (which is based on Allagan tech) was reverse engineered to instead of push Dalamud into it's orbit with the larger moon, to bring it back into the gravitational pull of Hydaelyn and ultimately closer with the Lunar Transmitter at Castrum Novum.
I wouldn't call the Garlean war on Primals as saving the world. Garlean magitek weaponry is based on Ceruleum, which is highly concentrated-refined aether (a major reason why Midgardsormr was killed in the explosion with the Agrias, the Ceruleum engines). Primals and their aether consumption go against the Garleans their need for aether to be used in their weaponry. Of course all this is assuming that Primals in other parts of the world are summoned in similar ways (I'd guess they are). Eorzea may be the strongest source of aether on the planet but I'd assume the aether flower of the planet has it's major rivers and tiny tributaries all around the planet...thus giving the Primals a source of 'life' in the other parts of the planet to be appear/be summoned.
Gaius was on a ship watching the Battle of Cartaneau Flats and I highly doubt he knew what was in Dalamud, but he might have had his suspicions. He was too far away to make an impact on Bahamut when the "Blinding White Light" appears. Sadly we won't know what happened after we were teleported away until we log in and see the CS for the first time. But it's most likely not Garlean involvement. After all Gaius was under no obligation to help the VIIth Legion escape their fate with Bahamut...the VIIth Legion was comprised almost entirely by brainwashed Ala Mhigans. The Garleans essentially only lost Ala Mhigans and magitek weaponry. Even though Nael van Darnus was lost prior.
With the arrival of the Elder Primals (not saying Odin was the one to stop Bahamut and not saying it wasn't) Gaius is bringing in the "big dog," Ultima Weapon...which of course I'm sure will be aptly named. The Garleans know what they're doing and know how to deal with the Primals, Nael van Darnus was happy to sit back and let us defeat them so he could harness the aether for Dalamud/Bahamut...Gaius isn't sitting around. He's now backed by 3 more Legatus (probably Legatus but we don't actually know for sure) and Ultima Weapon, along with his highly trained XIVth Legion comprised of Garlean troops and magitek weaponry...whom also still hold Ala Mhigo hostage. Gaius is also the most loyal to the Emperor (having killed 8 or so would be usurpers for Solus Zos Galvus) and it's his turn to deal with the city-states, Primals, and the traitors of the Garlond Ironworks.