The problem is not disputes over the authenticity of the signature. The problem is disputes over what that signature is connected to, i.e. disputes of the form: "Yes, that is my signature, but that is not the document I signed, this over here is the document that I signed." That is the reason that when you sign a physical document you sign the actual document, and not a blank piece of paper that you then attach to the document with a paper clip.
That's what cryptographic signatures are for ("digital signature" vs "electronic signature"). Hash the document and then sign the hash, to establish probabilistic certainty as to exactly what they signed.
