I highly doubt that not everyone who was/is culturally Garlean can't use magic. In the current-day Garlean homeland, a full third of the population isn't even Garlean by race, but is Garlean by culture and always has been. And Garlemald has always prized magic users no matter where they come from. I can seen non-Garlean-race Garleans being highly encouraged to take up the magical arts. Also, there's good reason the Garleans are the ones who invented firearms, and from the sound of it, the invention of firearms predates the invention of magiteck. So I have a feeling that once the Garleans got to northern Ilsabard, they doubled down on figuring out non-aetheric ways of keeping up with mages to supplement the mages they did have.

The thing is, even in aether-heavy societies like Eorzea, mages are not the majority of fighting forces. They are still the minority compared to martial fighters. Garlemald just has the percentage of the population who can use magic even more heavily skewed then usual. It's less, magic vs. no magic and more, many mages vs fewer mages. But where the fewer mages have a lot more infantry support. There's reasons Garlemald has the most organized military we've come across...