The Search API is acting as a database of a collection of webpages where the collection layer is the copyrightable thing. This is analogous to Google being able to copyright its search engine without having copyrights to all of the underlying content within the search corpus.
If your quibble is about the crawling / scraping, I'm not going to argue. It's easy for mass scraping to fall into unethical / illegal territory and I think that is the original sin of the AI foundation models.
Databases are copyrightable whether their content are facts (one example of content) or other data. I used the word facts because I knew from memory that was protected in US IP law.
A full copy of the indexable web isn't a creative work at all. It isn't a collection that has been curated, it is just the whole thing.
It is just like if someone scoured the internet for images and sold them in a book as a collection. They couldn’t do that without rights to those images.
Now imagine that the book “author” says you can’t copy the images from the book because they claim to own those images, without any copyright of any kind.
That is what they are trying to do. But with a different kind of copyrighted work.