I like the idea, but not as much as I like having your FC mates living in a neighborhood with you on beach-side residences.

In terms of this actually happening, I don't see someone actually flying the ship but rather an NPC that is selected (similar to selecting a companion) and the ship being instanced.
From this, I have some ideas (sorry if its been stated. I didn't read through the whole thread)

1) If it is instanced, there can be an additional option in the menu called called "Call Airship". Once your Free Company has bought and built an airship, players can call it to them (with maybe a 5-10 minute wait) and then they are then transported to the ship flying above.
2) This can make the airship become a central hub for meeting prior to raids and such (even take away some of the congestion in Ul'dah)
3) The airship can drop players off at a certain camp without having to have all of the players in the same party. This would make it possible for traveling as a Free Company rather than traveling as a party.

The reason I don't see the airship being pilotable is because I can't imagine how the devs could possibly make a bunch of airships flying around in the sky work, let alone make flying work while dividing the map into smaller zones.