Your second sentence contradicts your first sentence.
> Publishers don't "sell their inventory", since everything a website puts up on the web is essentially free.
They have spots on their web pages for advertisements that they sell to Google or other ad networks. Brave unilaterally takes those spots from the publisher for a price the publisher never agreed to.
> The publishers chose Google as their middle-man to make money with ads.
That's the point. They sold that inventory to Google, not to Brave.
No it's not, at least in the EU as of last month's copyright reform. Meaning aggregator and search sites need to pay royalties for syndicating significant portions to publishers. That it's technically possible to scrape content doesn't mean scraping doesn't run afoul of copyright legislation and press norms such as proper attribution. You could also technically "scrape" written books; yet re-publishing your own book copies isn't considered legal.
Publishers don't "sell their inventory", since everything a website puts up on the web is essentially free. Consequently there isn't a licence involved, in contrast to your movie example. The publishers chose Google as their middle-man to make money with ads.