Earlier quoted context omitted.
A publisher publishes. Once something is published, once something is public, the control a publisher has over the published thing is limited. For example, a publisher can not choose who reads a book after it is sold, who reads an article after it got printed. It is not a given at all that a publisher should have any say about being indexed. The search engine relies on a public fact - X wrote Y. That's legal (limitat…
A search engine index is an economic exchange between the website and the publisher. To massively (over)simplify the argument to its essence (and ignore other important points): the publisher goes through the trouble and expense of creating the content The publisher then allows its content to be copied by a search engine only because being shown in search results gets it traffic back. The traffic it gets in return ha…
A search engine index is a search engine index. It can have an economic impact, but it can not be an economic exchange, since it is a technical artifact.
Though I think I understand what you are trying to say - this is also a commercial relationship where both sides can profit. You are free to interpret the relationship between publishers and search engines with such a capitalist lense, that does not mean those mechanisms govern the actual rules. That a publisher is happy with what happens here is of no real concern. If any rules apply we are talking copyright, maybe media law, where happy is not a relevant category (ok, that of course can matter, it wouldn't here if the search engine uses a right).
I did not touch the LLM training data in my comment, as I did not read up on what Brave is really doing there. If Brave were really to sell complete texts from others, that would not be legal under copyright laws I'm familiar with, so I kinda doubt they do that.