Not legal advice, not an IP lawyer, and not your lawyer but…
Is your fic copyrighted?
Yes, your original contributions are copyrighted. You just don't own the elements that were created by the creator of the underlying original work.
Why are people allowed to remix?
This is largely cultural and also an administrability nightmare.
Culturally, fanwork is transformative and collaborative. There is a long history of remix culture that is part of the ethos of fandom.
On the administrability front, AO3 simply doesn't have the time or means to adjudicate issues of copyright infringement outside of clear infringement (wholesale, non-transformative copying). When someone makes an allegation of copyright infringement, unless the accuser is willing to take the accused to court, there is not legal remedy for the violation. AO3 is not a court and they are not set up to create reader trials to determine if a violation occurred and, if so, whether fair use would or would not apply.
You can't copyright concepts, only the expression of them (so the actual execution). So, unless someone literally copy and pastes your work, in the case of full re-writes it can be difficult to even prove the underlying copyright violation since you'd have to prove that they not only read your work but also that it was your work that they were copying (subconsciously or otherwise).
This a nightmare already in courts, so there's just no way AO3 could fairly administer such a scheme. Wholesale banning (or requiring permission for) remixing would lead to AO3 having to adjudicate alleged instances of unacknowledged remixing and determine if something is a remix of if it's just someone who had the same idea/concept applied to the same characters who would, informed by their canonical personalities and situations, generally react in a predictable way.
Why are people allowed to download your work?
It is part of AO3's mission to promote accessibility, preservation, and user control over how they consume content. When you upload to AO3, you are agreeing to allow this to happen. If you do not want your fic to be downloaded without it being a copyright violation, you would need to host your fic on a site where there is an agreement between site users that downloading a fic for personal use is an approved use. You would also then have to personally pursue anyone you believe to have downloaded your fic for violating your copyright.
How is AI scraping different?
When an AI org scrapes your fic, they are explicitly not downloading it for personal use, which violates the agreement that users of the site made with AO3. Even they don't use your work to create derivative works, this is a copyright violation (unauthorized copying). (There's a sub-issue of whether the work is stored and could be reproduced on request which would be (unauthorized publication and dissemination).)
The fair use issue is undecided. Fair use is an affirmative defense to copyright violations, meaning that when someone makes the argument of fair use, they are admitting to the underlying copyright violation. Fair use just means you can't be punished for it. That's why even if AI scraping and training is fair use, it's still a DMCA violation.
Why do you have to file the DMCA claim?
DMCA takedown notices must be filed by the copyright owner (you) or an authorized agent/service (simply not feasible for AO3 to be this service since it would require individual authorization from each author and require AO3 to keep track of scraping and then determine on a fic-by-fic basis whether they need to and are authorized to submit the notice).
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