@ryanc Plus there's a ton of ways to fingerprint a browser (or get the same information from a different source, like navigator.appVersion instead of the User-Agent header).
To continue on the clown analogy, even if you'd get a group of friends to wear random makeup… you're probably not changing all your recognisable traits, after it's possible to recognise friends with different makeup/haircut/… or even full on cosplay.