The big difference between Momo and Carrot as leaders of their nations is that Momo wants to lead, he time and again stated that even though the task itself is much too big for him, whereas Carrot never showed interest in leading.
Adding the whole "you gara are carrying Pedro's will" doesn't make a lot of sense to me, because I don't associate that with being the next leader of Wano.
Pedro set out to find more poneglyphes, so that he could help the person following in Rogers's footsteps and thus help bring the dawn.
But then again that whole "dawn concept" is so vague it could be many things.
