Some of the modern Aetherytes in the city-states were created by Sharlayans:
The aetherytes currently found in each of the city-states were reconstructed following the Calamity by Sharlayan engineers. These engineers were the only ones who possessed the skill and knowledge necessary to do so, and so they were richly compensated for their services.
I wonder if the Sharlayans make aetherytes for the nations outside of Eorzea?

There needs to be concentrations of aether for a aetheryte to be made
Aetherytes cannot be constructed just anywhere; rather, they can only be constructed in locations rich with aether, such as the intersections of aetherial currents. Like an oasis in a desert, these locations are often teeming with life. Settlements are commonly established near them, and so it can be reasonably assumed that most villages and cities are ideal locations for aetherytes.
It's possible that aetheryte technology might have even predated the Allagans
The Allagans possessed aetheryte technologies as well. However, it is unknown whether they developed their methods independently or if they expanded upon those of older civilizations.
http://gamerescape.com/2015/10/28/lo...-lore-lowdown/