You give up an element of this anonymity and repudiablity when using a mail-in ballot system instead of a truly anonymous in-person voting system. The votes are still tallied anonymously, and it's impossible to post-hoc validate who you voted for, so it serves the anonymity purposes since you can't check whether your investment in buying votes was successful.

Every state that does mail-in voting at all requires a signature. Alabama, South Carolina, North Carolina, and Missouri also require a witness -- Missisipi requires a notary.