Gameplaywise the gil needs to be removed from the game.

Lorewise you're paying a fee because you're using the aethryte-net - I'm paying the train-company aswell when I'm taking a train to my house after all!

Dramawise I dont think automatic deposits into the FCs gil-bank is a just a bad idea in general, seeing how the FC-leader can cut everyone of from that pool and just take the gil for themself. If someone wants to donate gil and is aware that anyone who has access to the money can in theory use that for themself, thats okay. But any thing that could turn an FC into a profit for the leader should be avoided.
If an FC wants an "income" they can have one by selling goods they bought with FC-credits (THATS the real FC-income that everyone is contributing to after all) or that the vessels brought back - or whatever the members put into the chest out of free will.