I am wondering about the viability of introducing an eatery/food stall on the island, as we have both produce and animals to use. As a foodie myself, I would love to see this implemented in some form.

You would be able order and consume food that the mammets make. (Maybe they can get a certain NPC over as the head chef) in order to get IS exclusive buffs. For example:

Lost Mutton Soup: "Double materials gotten via gathering for 30 mins"
Fried Chocobo Rice: "Double running and swimming speed for 30 mins"
Chops of Paradise: "Increase probability of catching rare animals by 25% for 30 mins"

I can at least see the additional gathering buff being an incentive. If the 7.0 visual overhaul in the future includes improvements to the food models, it would be a bonus too.
Welcoming further discussion on this.