I'm not an US citizen, I've never voted in the US, I have no idea how your shit works or not.
But I know how my country's system works, and it's super fucking easy. On election day, I got to my voting place for my voting district. Each district contains a couple of thousand voters. For each election I'm participating in - national, regional, local - I place one piece of paper with the name of the party I'm voting for in an envelope, secretly.
I then go to the election officials, identify myself, and they look me up in their voter roll, which is simply a printout of the couple of thousand voters who should be voting in this place. We all check that I'm in there, and that I'm not already marked as having voted. Then they strikethrough my name in the roll, and put my envelopes in the corresponding urn. I am now certain that my secret vote went into the right urn, and everyone else can be certain that I've only voted once.
Once the voting ends in the evening, election officials in each voting place opens their urns, and opens each envelope, removing the vote inside. Then they simply sort and count the individual pieces of paper that are the votes. You count the total number of votes, you count the total number of voters crossed out in the voter roll, and make sure it all matches up. And this gives you a preliminary vote count for each political party in each district, so you only need to report a couple of numbers to the election authority.
Anyone can watch this process! I can go and watch it. You can come and watch! And it's so simple to count pieces of paper that anyone can do it. And with observers from the various political parties, you can't cheat.
After the preliminary counting, you take all the votes and your voter roll and put it in a sealed box and send it to the regional voting authority where they do a final control count, and archive the votes for history.
As for me and my vote, I can check my district on the website of the election authority: https://data.val.se/val/val2018/slutresultat/R/valdistrikt/0...
I am one of the 1160 voters out of the 1363 eligible, I can see that the party I voted for got at least one vote, so I can be fairly certain that my vote was counted, while what I voted for is still a secret.
Compromising this system is incredibly hard, because it is parallellized and distributed. And the paper trail will always be there. Somewhere in a vault, there are 6,535,271 pieces of paper representing the 2018 vote for the national parliament. You can't accidentally erase that or lose that.