That’s correct, the public Search API is (and always has been) limited to a 6-9 day window and is not guaranteed to be a complete index. The timelines only return a maximum of 3200 Tweets and there’s no way to increase that.
For a longer timescale, we’d recommend Gnip, which is our paid product for access to the whole Twitter archive. We also have a limited Data Grants program for researchers, but that is currently closed to applications.