The ability to make your own vote public is a bug, not a feature: It allows people to demand you to show them that you voted "correctly", whether for straight out vote sales, or plain old intimidation from a group. A pastor/union boss that can be sure that a bunch of people really went to vote, and voted they way they want, is pretty powerful. At some point you make the "voluntary" part of giving away your hash to not be so voluntary, as only traitors would keep it quiet.

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.

Example: I usually do not vote in the party primaries in my state, because the party that primary voters selected is public. Mostly, I don't want to get tagged by Big Data as having a political affiliation.

These combine with "at will" employment such that I could be fired for voting in the "wrong" primary for my company, with no laws broken. I'm not saying it has ever happened, but it could. Murphy's Law provides the rest.

In the general election, the only public information is whether I voted or not. So I vote as I please. I can still be fired for wearing only the minimum required amount of flair, but at least that's not interfering with democracy.

Public votes can also be sold. If you say you'll pay me $X if I vote for Y, making the Y vote public means they buyer can know that they got what they paid for.

So would it be impossible to make a system that would allow a person to lie if needed and also prove that their vote was wrongly assigned if needed? I've seen cryptographic solutions to things that seem as impossible initially (like having a system verify you know your password without ever telling the system your password and without the system ever knowing what your password is), so I wouldn't be so quick to write off a potential solution to this problem.

How do you verify that the machine is running the same source code?

Its being debated whether ballot selfies should be allowed or not, because it is protected by free speech or because it violates current laws about voter bullying. So a system that allows third-party verification of an individuals vote is probably against the law, at least currently.