It would open the door to union stewards, as just one example, demanding a copy of who their 'people' voted for.
A person's vote will ALWAYS remain between him and God (and anyone else he choses to tell.)
Hard copies of votes cast create FAR more problems than solutions...
A hard copy doesn't mean it can't be anonymous. It just means there is a back up record of the actual vote that can be audited. You look at your copy, confirm it is accurate, and go on your way. Later, the electronic count can be compared to the hard copy count, both of number of votes and the number of voters. Double-entry bookkeeping is a good thing.