In many environments, secret votes and public votes lead to different outcomes: requests to make regularly public votes secret is not that uncommon in elections for private groups: Everyone knows that there's a big percentage of people that want to vote in an unpopular way, and suddenly either an old leader is ousted, or a very corrupt choice is made.
So no, the ability to voluntarily make your own vote public and verifiable is a dealbreaker for elections in a modern state.