Just tried to disable User-Agent in Firefox by setting general.useragent.override to an empty string and Cloudflare captcha become impenetrable. Cloudflare actively blocks attempts to improve privacy :(
You don't want that to be an empty string - you would be one of the few people in the world with that value and thus easy to identify. You instead want that string to be exactly the same as everyone else in the world (no matter what the real User-Agent really is). there should be about 50 different contents of the entire header possible for everyone in the world.
