While I don't doubt dataskydd's good intentions, their advice about referrers is a sign that we live in Clown World.
Yes, your browser's tendency to provide a referrer might well give away information you would prefer it didn't. Unfortunately for you, the browser vendors have chosen to provide browsers that do that.
In a parallel universe it would be obvious that this is a problem (among many) for the browser vendors to address. In Clown World, you are supposed to rely on each and every site providing a special response header.