[Based on “canon” but not strictly. Used for local stories.]
Most citizens of Eorzea have some potential aptitude in the use of aether and magic. Just like all skills, it varies from person to person. There are very few races which lack any ability to manipulate either, such as the Garleans.
However, while aetherytes are widely available across Eorzea and the rest of Hydalean, not everyone is capable of using them. Only those individuals with a high aptitude in aetheric manipulation are capable of attuning to them and then teleporting across long distances like this. This is why most people in the stories will travel by using traditional means despite the availability of an aetheryte.
This is also related to inability to teleport large items to an attuned aetheryte. There are theories, such as an aether bubble which only fits around the user. This may explain why the user floats before teleporting.
