So for how many of your school exams do you think someone actually authenticated your signature? If it is never checked, it is reasonable to argue that it isn't being used to verify anything. Also, on things like exams, it is being used to encourage the test takers to take the honesty pledge seriously (and again, if someone signs a wrong name, it shows deceitful intent).

Identity is much more slippery than you are making it out to be. A signature doesn't prove it. An authentic government document doesn't prove it (it just verifies that you went through that particular identity verfication process).