What if the (rotating) role of representative is explicitly of lower status, lower authority, than the other people in the group? Not leader; lower-status representative. Administrative servant to the group.
Well, honestly that sounds like hell on earth for the unlucky person. I think that would disincentivize inter group cooperation.
That's basically what my team's administrator does; in the old days the role would have been called "secretary". He doesn't seem to be living through a "hell on earth" and has a cheerful, chirpy demeanor, but I'll double-check tomorrow that he's not about to top himself.