From what I can gather, the Consulate is the ruling government of Kaladesh that is some weird combo of "meritocracy", "technocratic government", "oligarchy". There are different tiers of the Consulate Government, and depending upon how well you performed you get promoted( Baan, Baral, Tezz). In fact, Tezz being given as much power as he has is a sign that he's made some friends in high places. Baan seems ignorant of Tezz's association with Bolas, so he's not pulling that kind of strings, but Baan did remark that Tezz has risen remarkably fast.
Pia's Beef(And arguably the beef of every Rebel) is that the Consulate believes they have the right to control who gets the Aether and how much they get. The Rebel Alliance disagrees and Chandra first showed her pyromancy when running a smuggling mission. However, Tezz ends up serving as the symbol of the corrupt consulate because since he seized power, the Aether flow has been constricted/the government has cracked down. Think Donald Trump and Racism/Sexism/Bigotry. It's hard to rebel against the systems that are in place, but the moment you have a face to it revolt becomes easier.
From the Artbook pictures that I've seen, it appears that the rebels stop rebelling once Tezzeret is kicked off and the Consulate finally decides to stop constricting Aether. The big "change" that the revolt brings is the freedom of Aether for the plane. I'd also not be surprised (supposing the art book is true) if Pia is given some role in the reformed Consulate as an olive branch to the rebellion.