Our current voting system has an emphasis on the secret ballot.
There are records of which polling place a person voted at,
and whether they showed up to vote, but (in theory) there's no way to tell which of the thousands of ballots from that polling place match to which individual voter.
But there is also no way for us the verify a valid and safe vote. Especially with our lively die bold machines. I'll take the thugs, give me a fair vote.
People from several parties, in a public, observed location. It's not hard; it works here in the UK, and it worked in the US before all this voting machine nonsense started.
There are records of which polling place a person voted at, and whether they showed up to vote, but (in theory) there's no way to tell which of the thousands of ballots from that polling place match to which individual voter.