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I find it difficult to wrap my head around how Privacy Pass actually works. It seems pretty involved. Kagi‘s implementor actually explained it over on hackernews, but that‘s only a summary.
Since it‘s an RFC standard, I don’t doubt that it works, but I hope someone smarter than me actually checks Kagi‘s open-sourced client code, which should be enough to ensure anonymity, according to the standard.
I unfortunately can’t really see how a browser could still be nice to use and properly resist fingerprinting.
The site https://amiunique.org/fingerprint tries to fingerprint your browser and lists the used attributes along with their uniqueness within their dataset. And while a browser could pretty reliably lie about its User Agent or Platform, it’s often just necessary for a modern website to know, for example, what your view-port’s resolution is or what kind of audio/video codecs your device supports. Going through my own results, I’d say combining these necessary data points is probably enough to identify me, even though I’m pretty privacy-conscious.
Maybe I’m overly pessimistic, but I think preventing fingerprinting would need a regulatory instead of a technical solution. Unfortunately that doesn’t seem very likely anytime soon.