I think you have that backwards? Midas' growing obsession with the Meteor project driving away Cid is what led to Gaius taking him in, which all happened after the occupation of Ala Mhigo and the Battle of Silvertear Skies. Gaius' and Cid's relationship was probably at it's best during this period, when he was like a second father to him, before the Bozja Incident caused Cid to defect from the Empire.it was so bad it soured Cid's relationship with Gaius and was a major motivating factor in his defection.
In regards to the Garleans, we have remarkably little of their history relating to the periods around their rise as an empire. We don't know the state of their nation or any conflicts they may have had with surrounding countries in Illsabard, and we don't know how exactly Solus rose to the position of emperor. As far as Emet goes, we know he did a number of things during his reign that are strangely at-odds with the Ascian goals for Garlemald, particularly backing down from invading Eorzea after Silvertear and shutting down the Meteor project for ten years after it destroyed Bozja. And his patronage of the arts rather than creating a climate of censorship devoted to propaganda, but maybe you can just call that a distracting side project.While we don't have that much lore on the Republic( SE we want a Garlic expansion T_T), I feel like what is the closest to the truth would be that the Ascians fostered the seeds of prejudice that already were due to the Garleans being mistreated by their neighbors and blowing it out of proportions by giving Garlemald a leader they worship nudging them in certain direction. Where Solus pointed, people followed because they agreed with him. And it wouldn't be the first time the people in this game or IRL followed a charismatic person whom they regarded as a savior of sorts.
Frankly, I'm pretty sure Solus being an Ascian was a relatively recent story decision, and it's hard to reconcile that with some of the other things we know and their overall plans.