Homomorphic encryption

Homomorphic encryption does not affect that problem... It's just down to policy. If the decryption authority “stays open” after the election and no longer insists on checking ballots to see if they have already been cast, then yeah, you can abuse the system to decrypt placed ballots.

If the keys are destroyed after a valid election, as one would expect, then there is no possibility for that.

One way to better ensure the keys are destroyed is to use secret-sharing schemes so that multiple parties that are adversaries would have to lie similarly about destroying the keys, then conspire to work together to decrypt ballots after the fact. But I hope you see that this is all chasing social problems that must be solved as a precondition to have fair elections in the first place.

Right, the protocol is essentially that you have a central server which supports in essence two SQL queries,

    UPDATE ballots 
    SET status = "burned" 
    WHERE contents = :ballot AND status = "unused"

which, if it succeeds, then sends the ballot to the decryption oracle with the private key, to be decrypted and sent back to the user; and

    UPDATE ballots
    SET status = "used"
      , voter_id = :voter
      , choice = :choice
    WHERE contents = :ballot 
      AND status = "unused"
      AND region = :region

which, if it succeeds, then sends back a confirmation that this user has been logged with that ballot and made that choice for that ballot.

If you allow people to access the decryption oracle without going through that first pathway, which simultaneously checks if the ballot was not spent and immediately spends it into the "burning" pathway, then either of those opens up the space to attacks which decrypt individual ballots. With that said, just about any auditing mechanism applied to the decryption oracle would be revealing the existence of those attacks anyway so you can still get a measure of security without this.

You can potentially even distribute the database (e.g. over a blockchain among several political parties), but as far as I can tell the decryption authority would still need to be centralized and could be a single-point of failure. (In this case it would be a program which is watching that blockchain and interacting with it via some “I publish a burned ballot onto the ledger after I think the blockchain has passed N blocks ahead of the ledger request to burn that ballot” algorithm, and nodes in the network need to reject requests to cast ballots that they think have been requested to be burnt.)

I don’t understand why you think these things are necessary or opposite.

You can have a system where everyone has a copy of the database. That is not hard, it just requires the separation of what a ballot means, from what is stored in the database. That is just these two-sided ballots with encrypted values on the right-hand-sides: so that the fact that I voted for #1 on my ballot does not tell those who hold the database who I voted for.

You can have a system where encrypted ballots are known by the people to have the values that they say they have. That is not hard, it just requires a challenge-response scheme. If I give you a box and claim there is a pony figurine inside, you can be suspicious: if I give you twenty thousand boxes and claim that they all have pony figurines inside of them, and you ask me to open ten thousand of them which you choose randomly, then for me to omit one pony I am facing a 50% detection rate, for two I am facing a 25% detection rate; to disenfranchise even 10 people from their ponies I will be caught in the act 99.9% of the time, and even then I can only disenfranchise 0.1% of the boxes.

So I can have great confidence that my vote was recorded for the first person on my ballot (I can see the database), and I can have great confidence that the first person on my ballot was Alice and not Bob or Carol (because they passed my challenge/response test).

You can also have a system where nobody can pay substantial sums of money for votes. That is also not hard, it just requires the things that users take home with them out of the voting booth to be easily forged, so that they cannot prove that they did not forge the thing.

Absolutely none of this requires homomorphic encryption; homomorphic encryption just streamlines some of the process around the decryption oracle: with HE tallying and anonymization happen outside of it, so that its internal structure simplifies drastically.