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 (limitations apply).

The position of Brave of not accepting to be blacklisted if not all search engines are blacklisted is a pragmatic one. It works against Google's search monopoly, but still gives them some legal coverage since the robots.txt is not completely ignored, in case that indeed matters somewhere. I think it's an elegant approach well suited for the current state of the web, one that serves the greater good. And Brave, but that's completely fine.

No, it's not.

But why do you believe so?