When I started using AI for serious work, I had the same requirements as for any other projects: local backup of my files. To achieve this, I decided to mirror Google AI Studio session files from Google Drive using Rclone.
To view them, I wrote a Google AI Studio HTML viewer. For Gemini CLI, I wrote a similar viewer for .gemini/tmp/*/chats/session*.json session files.
I find having these files locally very useful for searching (Ctrl+F); you might too. You can also copy any part as markdown or html.