Kagi is a privacy-focused search engine whose incentives are aligned with its users, and I've really enjoyed using it the past few years. the extension works well to allow always-on sign in to the search engine in private windows.

Finally switched from Google – and I’m never going back

I’ve effectively switched from Google Search to Kagi Search and I absolutely love it. After trying the $10/month plan, I moved to the $5/month Basic plan and have been using it happily for two months now.

The Firefox extension makes the experience seamless: it works perfectly in private windows without requiring me to log in, and the instant page summary feature is incredibly useful. The extension is open source, lightweight, and just does what it says without any bloat or tracking.

If you’re tired of algorithm-driven, ad-cluttered search results and want a clean, fast, privacy-respecting alternative, Kagi is worth every cent. Five stars without hesitation.

I believe the main feature is that it will sign you in when searching in private tabs, which works well.

You have to manually change your search engine to Kagi on Android at the moment. I guess that's a restriction to Firefox on Android.