Brave Browser includes several convenience features, mostly linked to the built-in content blocker. One of the features deals with cookie prompts automatically.
This works well most of the time. Whenever a cookie prompt appears, it is handled immediately by the browser in the most privacy-respecting manner.
Announced in 2022, it has been a feature of the browser ever since and is enabled for all users since mid-2023.
Here is how Brave describes the feature:
One approach (which Brave uses) is to block cookie banners, and to hide and to modify pages to remove any additional annoyance such systems include (such as overlays, preventing scrolling, etc.). Other Web-privacy tools (such as uBlock Origin) can be configured to use this same approach. This approach provides the strongest privacy guarantees: it doesn’t require trusting that the cookie consent systems will respect your choice, and prevents your browser from needing to communicate with consent-tracking systems at all.
Brave blocks cookie banners and deals with any nuisances related to them, including overlays.
However, while that works well most of the time, you may come upon websites that do not seem to work at all. You do not see a cookie prompt, but you can’t interact with the site. No scrolling, clicking on links, copying text. Nothing works.
This may be related to the script that deals with the cookie banners. It is possible that the site changed its code or that Brave does not support the specific cookie prompt the site uses. Brave’s solution relies on EasyList, a community managed list, which means that it works only when sites use one of the identifiers of the list.
Reloading does not resolve the issue, as the site will get stuck again. In fact, the only option that is working is to allow the cookie banner script to display and deal with it manually.
Sidenote: You could dig in the code and find the references to block them manually. May be worth a short, if you visit the site regularly.
The quick solution: Reject everything, deal with the cookie banner manually, and enable the script-blocking of the browser again for the site.
Granted, this is not ideal, as you allow the site to run all of its scripts and such. You could try and allow certain scripts to run only, which is the better approach, if you have the time to adjust this manually, as you will limit scripts this way.
If you are in a hurry though, you may use the method described above to quickly gain access to the site.
Once you have dealt with the cookie banner script, you should be able to use the site in question normally.

Sometimes turning on “Block scripts” in the Brave Shield works too.
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