As others have said, this is exactly how you can start buying votes. Pull any book about cryptographic voting off the shelf and this is treated as a design flaw.
Jim is not a nice man and is very demanding and abusive and goes to vote. He wants his wife to vote for X, but he suspects that she will vote for Y. Today, she can vote for X or Y and tell her husband that she voted for Y. He has no way to prove her wrong. With receipts, he can tell her, on no uncertain terms, that she is to show him her receipt afterwards. If she doesn't produce a receipt showing she votes for X, he'll beat her. With receipts she does not have a free vote, without receipts she has a free vote.
I want to buy your vote, and will pay you €10 to vote for X. Today, you can vote for who ever you want, and tell me you voted for X and demand payment. I have no way to prove you wrong. Hence vote buying is hard to do, because you can't know if you're actually buying votes. With receipts, I can ask to see your receipt, and only pay out if you vote for X. With receipts, vote buying now becomes an actual thing that I can do.
Yes, no system is perfect, and the current system does have that flaw. But "record yourself marking the paper and putting it in the box" doesn't scale. If lots of people in an area do that, someone'll figure out what's going on.
Because if you can prove to a third-party how you voted, you can sell your vote, or coerce someone else to vote a certain way.
It's one of the biggest challenges in crypto-voting systems. And there are some solutions. But we need to hammer down on the "silly people, why not just print a receipt" idea really hard.
If the receipts are anonymous, they can't prove how you voted, only that you were able to obtain a receipt. So what if we increase the supply of receipts enough to destroy the value of an individual receipt? For example, voting machines could drop duplicate receipts in a bucket that voters have access to.
But I still would like to be able to have it proven to me that the way I voted matches what the system records. To me, that seems more important, but I also cannot say that it's more important than your points. Hm.
