I came across a post on the Japanese forums that I thought was pretty interesting. The OP mentioned wanting the fairy to actually do something other than fly when out of combat. For instance, sit/stand on your shoulder like the job art shows or sing and dance in the air.

These ideas are amazing to me because they already exist.....in the form of minions, yet they've never been implemented for the SCH's fairy. For instance, Midgardsomr can land on your shoulder or head depending on your race. Moogle minions will dance together and Mandragora can take turns dancing.

There seems to be a vital lack of fairy interaction in the game and it's painful since they are supposed to be your partners, yet you rarely acknowledge them and they rarely acknowledge you. They just sort of follow you and pop out a few heals every now and then.

For background, the original post is from 2014 and it received a reply from who I assume is a developer (it had a crown on the question which I think means official response) who said that it wasn't planned to be added and neither were new fairies planned to be added. Well a new fairy has been added this expac (sort of) so maybe it's time to revisit this idea.