That's not Twitter's job. There are people who are supposed to do this.
Whether it’s good that they have become a news source or not is a valid question. But right now, they are, and I think preserving the president’s tweets seems like a pretty easily defensible decision. Even if it is also (or primarily) motivated by their own business interests.
If the answer is yes then Twitter isn’t performing the civil service it claims as it is not an archival medium.