Hello @rolak
i know you expert but just in case you missed something
- make sure that hangout extension enabled here
brave://settings/extensions - did you check it on private mode
a) if private mode work then maybe the issue with one of your extension so try to disable all of them and if it work then re enable them back one by one til you get the one that cause the issue
b) if extension was not the issue then try to clear the cache and cookies
if all does not work then start brave using this flag --user-data-dir
check this for more info about that trick
hope that help and have a nice day 