The problem could be solved with the setting simply not removing the cookies of the allowed websites of the user.
It is a big problem as certain websites like https://search.brave.com/ could have settings within the website saved and thus it would require users to manage the settings the same way as previous sessions on every session.
And it seems that the issue has been suggested to be taken care of for a long time, yet there haven’t any been any steps taken forward with it.
Brave version: Version 1.46.144 Windows version: 21H2
Hello again @daze, by any chance do you have the option under settings Menu --> History --> Clear browsing data --> On Exit enabled? If so, please disable it and let us know if it works for you. Regards.