You don't sign the letter because signatures have a way of magically[0] teleporting themselves onto documents that you've never seen.
0 - Using such fancy Merlin's wands of "Photoshop" or "scanning and then MS Paint copy/paste" or just "photocopy, cut with scissors, paste with tape, photocopy again."
That is highly illegal. The FCRA and related laws are enforced by the FTC, CFPB, and debtors' lawyers. However, forgery is a felony in all 50 states. If they forge your signature and mail it to you from another state, it becomes a federal crime. A creditor would have to be really stupid to try something like that.
0 - Using such fancy Merlin's wands of "Photoshop" or "scanning and then MS Paint copy/paste" or just "photocopy, cut with scissors, paste with tape, photocopy again."