The role of dog is one of criticism. As with dominance, criticism is a loaded and twisted word. It is largely seen as negative (and thus dogs are often thought less of than is fair), but good criticism can make a project work. It can decide between ideas that are crazy and won't work and ideas that are crazy but will work, a distinction cats are by nature hard pressed to make.
There's a bit more to it than that though: a dog can say why something is a bad idea, and a dog can say this not only about potential actions but about the world as it is. A dog is a problem poser.
A dog can say things like "our OS lacks a robust file-system" or "it's really crap that poor people are starving" or "psychology is very focused on disorders, we haven't really looked at what makes people well" or "copyright is broken for the following reasons...". A dog leader can then choose amongst the best solutions that are proposed by his or her cat minions.
Every form of government we have tried has had cat leaders. Monarchy. Democracy. Fascism. Communism. Every form of government we have tried has been, well, really crap. The vision of a single person is not sufficient to run a country, or a multinational corporation or NGO, or a religion. It seems the one thing we haven't given tried is dog leadership. I reckon we should give it a go. Yes or no?