No, the Search API isn't claiming it is taking over a copyright of the original crawled webpage. Quit trying to strawman this thread.
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.