Here's some canonical evidence as to why Pinky and The Brain deserve to rule the world.
In A Pinky and The Brain Christmas and Brain's Night Off, Brain is a vocal supporter of worker's rights. "Is that the way you treat your valued employees? No wonder you can't find good service anymore, if every time a clerk tries to help a customer she gets yelled at! [Turns to the toy store employees] You people shouldn't have to put up with this. You ought to organize! Form a union! Stand up to your boss!" (Brain's Night Off)
Branching off from that, the whole point of Brain's Night Off is that Brain decides to spend an evening not trying to take over the world and ends up standing up to several injustices when he and Pinky were supposed to have fun. But unbeknownst to Brain, the people admire him for standing up for what's right and they rally behind him being their leader.
Brain has stated in multiple episodes and in Wakko's Wish that he wants to rule the world because he wants to make the world a better place for the little guy. In Megalomaniacs Anonymous, Brain tries to give up world domination for his own sake, but after he sees Pinky get assaulted by some jerks, he declares "If I ruled the world, people wouldn't step all over the little guy!"
In A Pinky and The Brain Christmas, Pinky states in his letter to Santa that Brain "only wants what's best for the world". Moments later, Brain does have control over the world for a moment, but he commands the hypnotic masses to have a Merry Christmas and spreads joy to the world.
In The Pink Candidate, Pinky is elected president and he manages to be a surprisingly competent leader. He loves completing his tasks, which include stopping wars. When Brain tries to use Pinky's position to enact his world domination plan, Pinky refuses to do so because it would go against everything he came to stand for (since Brain taught him the principles of democracy during the campaign). When Brain's past plans are exposed, Pinky takes full accountability for everything that goes on in his administration and admits to the Senate that it was his fault, and Ted Kennedy praises Pinky's courage and honesty (before he impeaches him from office).
In the Freakazoid episode, Freakazoid is History, Freakazoid ends up changing the course of history and finds that the world has become a better place. At the end of the episode, it's revealed that The Brain is the president.
In It's Only a Paper World, the mice build a 1:1 replica of Earth out of papier-mache, complete with biodiversity and architecture, and lure everyone to Chia Earth with free T-shirts, resulting in everyone becoming much happier. The mice also inadvertently save the global populace when a meteor hits the actual Earth.
In The Brainwashed Trilogy, the mice save the world from a nefarious scheme that dumbed down humanity.
Overall, Brain has an ego, but he truly does want to improve the world and Pinky is a kind and honest mouse and together they'd probably bring about world peace.
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