Sorry, Del, hate to disagree with you, but Carrot (was) meant for KINGSHIP not PATRICIANSHIP.
Besides his looks, his charisma is his only gift. He can only have influence over people when he is confronting them directly. Because he truly believes that everyone is, at heart, a good person, he is not someone who understands the strings that need to be pulled behind the scenes to maintain and wield true political power. In the short term, he might be able to get certain groups (like trolls and dwarves) to agree with him on certain issues, but, against AM's conspiratorial classes, he wouldn't stand a chance if he tried to do anything that wasn't in their best interests. He knows his limitations as a potential ruler, which is why he has never claimed the throne for himself.
Vimes also lacks Vetinari's Machiavellian skills. He is able to succeed at diplomacy simply by being honest and leveraging the Discworld's need of someone like him to bring the blunt truth to the matter. He's certainly succeeding in his diplomatic efforts (otherwise, he wouldn't have been assigned this role in Monstroust Regiment or serving as mediator in the Troll/Dwarf peace negotiations), but he's not a politician at heart (although he probably likes it more than he thinks).
Not that we will ever see a new patrician in the series, but if one were I'd still conjecture that Vetinari is grooming Moist von Lipwig to be the new Patrician for the Century of the Anchovy. Like Carrot, Moist knows how to manipulate people without making them aware they're being manipulated. Moist also understands the political and economic undercurrents that drive the city and can exploit them to his advantages. something Carrot doesn't have the intellect for and Vimes doesn't have the stomach for. The one thing Moist doesn't have is the ruthlessness of a Tyrant that's needed to do the bloody work when required (which is what Vetinari is constantly testing him to see if he does have the stomach for it). But Moist does have a strong sense of what gets the wheels of commerce moving--the most critically important attribute for the guild members who are most responsible for appointing a patrician. And, perhaps, the patrician of the future needs to break with the tyrannical ways of the past. After all, Vetinari is a very different kind of leader than the stupid, single-minded, violent idiots who preceded him. Who says that Moist might not succeed in his own way in the role?
J-I-B