I'm able to reproduce this on a fresh install of Bazzite on bare metal, and this has happened on Fedora as well in the past. Here is exactly what occured:
- Installed Brave via Flatpkak
- Enabled sync, which synced all extensions from my other computer
- Brave Search is still the default search engine
- Search something on Kagi.com
- Kagi is still not an option in the Brave search engine settings
Unfortunately since I'm using Bazzite, the best and most reliable way of installing Brave is via Flatpak. My only other option is utilizing Distrobox to create an Ubuntu container or something and installing Brave on that. I'd really prefer not to do that.
The extension should take control of the search engine setting entirely, and force it to be Kagi all the time, even in private windows.