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I’ve temporarily fixed this for myself. In my emails, I found a link to apply my original key. (Subject: Your Tabs Outliner Paid Mode License Key). I clicked the link, and it activated without the need for google. That has allowed backups at least, which is what I mostly care about. If you really want to get fully activated, find the extension folder, copy it somewhere else, remove _manifest from it, and load it as an unpacked extension in developer mode. Edit background.js, find function isEmailPemissionPresent, and replace the contents with return true. Find function getIdentityEmailNonInteractive and replace the contents with
var userInfo = {email:"[your email]"}; continue_callback(userInfo);

For those that found this topic first via search, progress has been made on this issue. Enable chrome sync in brave's settings.
https://github.com/brave/brave-core/pull/5616

https://github.com/brave/brave-browser/issues/5197

Just bought the paid version. This is a major request for me. It seems when using the chrome "restore windows on exit" feature (not after improper shut down), the windows still get added as crashed sessions. Very annoying.