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 has value, and the publisher is happy for this arrangement to continue as long as the value of the traffic is more than the cost of producing and serving the content.
Brave offering a "license", for its own financial benefit, to "allow" others to use the content for LLM training gives zero benefit to the original publisher. This is why I use words like "sleazy" to describe Brave's position.
This argument applies to Google and Microsoft. Right now both are failing at citing sources in their generative AI search results. That is terrible and I hope it's fixed soon, as otherwise they're being sleazy scrapers as much as Brave is.
Finally, I wholeheartedly disagree they what Brave is doing is for the "greater good". The fact they charge extra for the "license" to use the content for LLM training shows that.