To my mind this the real issue is the FIA not implementing the full year gardening leave when their employees are poached by teams.
They are only doing so because a Swiss court ordered the FIA to cease making gardening leaves longer than 3 months. This is not something a gentlemen's rule can override. The FIA is no more pleased with this than you are, from what I understand - the gardening leave rule acted as a deterrant for anyone wishing to leave the organisation. They were especially keen to keep Laurent, but Ferrari "outbid" the FIA.
It is Ferrari, not the FIA, which broke the "gentlemen's agreement", though it is not clear if a Swiss court would be impressed if it thought Ferrari was trying to enforce by proxy the agreement that had previously been explicit.
If the FIA had used French contracts, I believe this would not have been a problem, however there are other reasons why the FIA prefers to use Swiss contracts for its workforce. (I'm not sure how many of those are practical and how many merely historical). Besides, it is not clear how well it would work if the FIA started transferring contracts into French law, when it would be fairly obvious it would be doing so merely to make things more difficult for its employees...
I think the FIA will have someone in the wings, though it is unlikely to be their first choice. As for everyone who nominated me, thank you very much - if you can convince my workplace to let me have the weekend off, I would certainly be prepared to have a go.