It makes no sense that their aren’t other branches of the Citadel outside of Oldtown right? or that the other pre-Conquest kingdoms wouldn’t have built rival institutions to check the power of the Reach? Wouldn’t the Westerlands and the Stormlands and the Dornish have been uneasy with all these dudes from Oldtown showing up and handling their medicine and private correspondence?

It is a bit odd, yes. But if you think that’s odd, what about the Faith of the Seven? 

You’d think that the religious leader of the Vale, as the oldest Andalized region in Westeros (where we don’t even know about a single named sept!), would have demanded pre-eminence vs. that upstart at the Starry Sept when the position of High Septon was created. For that matter, why wasn’t there a schism or some kind of controversy when Oldtown suddenly lost its place to King’s Landing? 

The Lannisters would certainly have built a Golden Sept in Lannisport just to make it more magnificent than everyone else’s and be constantly bribing the Most Devout to get a Westerman elected as High Septon. And there’s no way that the Septons of Dorne aren’t going to have adapted their practices to suit Dornish customs, or haven’t picked up Dornish attitudes to the Reachermen, even if just for evangelical purposes?  

For that matter, given the Iron Islands’ contentious relationship with the Faith, why don’t the mainlanders consider the Ironborn heathens? And given the wars between the Vale and the North, you’d think Valemen of the Faith would be more anti-Old Gods than the rest of the South. Likewise, given its history, how does the Riverlands not have syncretic elements from the Old Gods and the Drowned God?