Well, if Azyz Lla can have an exotic Aetheryte design that's never used elsewhere, I don't see why Garlemald couldn't have it's own model. Also... I get the feeling Aetherytes would be the kind of thing the Allagans would manufacture...

One of the things about Garlemald that often gets overlooked when discussing it is that there has always been a sizable minority of non-Garlean races who have always been Garlean culturally. As in, with full citizenship, rights, etc. The lore book puts 30% of Garlemald's non-conquered population (not the population of the provinces) at being non-Garlean racially. If there's always been that many non-Garleans living in the Garlean homeland, I could see them having an Aetheryte system in place for them. Especially since Garlemald is so far north that everything we know about it makes it seem like Garlemald has the same climate as Siberia. Getting around in winter in those conditions would be way easier (and safer) to do with Aetherytes.

The other thing to consider is that magitek teleportation has probably only been around as long as magitek itself has. Garleans would still have needed to get around their Republic before that. Given how prized mages are too them, I could see them having Aetherytes specifically for the few mages they do have.