So here’s my thinking: there’s something really weird about the way that the WOIAF has the Maesters be a pre-Andal instituion but goes back and forth on the First Men having a written language.
So here’s my thinking. I think the runic writing of the First Men was quite complicated and difficult to learn, requring one to learn thousands and thousands of easy-to-confuse runes – and that the pre-Andal maesters thus relied as much more on memorizing oral traditions, similar to the traditions of the Celtic druids and bards.

And then come the Andals, but instead of overthrowing the system and burning the Citadel, they get incorporated into the power structure of the Reach and Oldtown more specifically. And so the maesters encounter these new Andal lords and knights, and there’s cultural sharing and intermingling going on.
Now, my headcanon is that the “Common
Tongue” is so named because it’s a relatively easy language to learn (atonal,
regular conjugations, no complicated system of cases and agreements,
straightforward grammar that doesn’t have the verb at the end of the sentence, etc.)
and a writing system that’s alphabetical rather than character-based, so it’s much easier to read and write and to teach people to read and write.
So early after the Andal incorporation into Oldtown, I think the maesters decided to adopt the Common Tongue and, with the help of the septons of the Faith, write down everything that had previously had been preserved orally, thus why “the tales we have now are the work of septons and maesters writing thousands of years after the fact,” and “the septons who first wrote them down took what details suited them and added others.”
Thus, we have a reason for the Maesters to compromise with the Faith.