I'd be highly skeptical, because I would hope someone who was a good enough member to get on staff in the first place would be mature enough to deal with the subject reasonably. That said, I wouldn't want to go into the discussion with "resign now or you'll be kicked out" perspective in the first place. Instead, I'd try to talk to them about whether they still cared about the place and if they still had a desire to contribute. If they explicitly didn't, then there isn't much left to discuss. Of course, depending on how big and immature the tantrum was, that by itself could warrant kicking them out.

I also don't like the idea of formal probationary periods, and would consider it a last resort (if at all). I think the first step should be "we'd like to see you around more. Please be more active or consider resigning so we can replace you." I'd try to keep it as non-adversarial as possible, because I've been on the rest of the staff's side of that process before in other contexts -- and when it starts being adversarial, that's when emotions get strained. If the staff member continued to be belligerently stubborn and feel entitled to the rank, regardless of activity, then we wouldn't have much of a choice. A lengthy probation period at that point might just serve to stoke the drama in the staff forum instead of just being done with it.

On the other hand, if they do seem sincerely contrite, then a probation period just shouldn't be necessary, and again makes the process needlessly adversarial. Just give them a second shot. Unless they've pulled the same trout before, why not trust them?

So basically: probation periods would seem to either draw out drama or are unnecessary, and either way make staff discussion combative.

But I really do hate that sort of entitlement attitude to a staff title.