EDIT:
Sorry, still waking up. Technically speaking, the Faith didn’t “put its seat in Oldtown.” Rather, what happened is that Septon Robeson was the most influential Septon in Westeros in his day, acting as regent of Oldtown, and became the first High Septon. Then the Starry Sept was built to honor him.
Then, “in the centuries that followed, Oldtown became the unquestioned center of the Faith.” In other words, as with the development of the papacy in Rome, it took a long time for one major center of worship to be seen as “the” center.