Discussed some on Zulip, I think people are in agreement that it doesn't necessarily have to be a member of the team in question to start a FCP.
That said, I'm not 100% sure what the restriction should be. Presumably not just anybody should be able to kick off an FCP (or maybe it's not a problem?). I guess also this could be a per-team decision on whether or not to allow this. (This is perhaps more trouble than its worth though, at least initially).
I can probably take this (barring it being more surprising than expected to implement), but won't be able to get started on it until this weekend.