Consider for example using blockchain for this. Every eligible citizen gets a "VoteCoin" from the voting officials and deposits it in the official "VoteWallet" of their party of choice. At some cutoff point no more transactions are allowed or considered.
Each voter could verify that their vote went to the right party, and the voting officials could easily verify the votes (no unknown VoteCoins for example).
However now everyone knows which party else everyone voted for...
In general the issue seems to be that if someone voting uses an electronic system they cannot rely on what that system reports back to them. It could be hacked to show whatever. And in order to remain anonymous the person voting has to be the one that verifies that their vote went to the right party.
1. https://en.wikipedia.org/wiki/End-to-end_auditable_voting_sy...
How do you prove that an electric system wasn't tampered with? How do you verify the voting machines actually run the verified code? How do you verify that some sub-component didn't hack the RAM?
Winning an election is too valuable and the risks to myriad for me to ever trust electronic voting. Especially as as you mentioned tampering with paper on a large scale will likely leak something went wrong, whereas electronic tampering might never be discovered and can be accomplished at large scale.