But this same lack of efficiency allows governments to make it harder to vote. Eliminate voting locations in regions where those who vote against your party are more likely. Move voting to days that the less-powerful can't take off from their jobs. Paper votes "go missing" because there's reason to believe the box of them contained too many that those in power didn't like.
Making voting hard makes the system ripe for abuse, and that's exactly what we see happen all the time.
What we need is a system that is efficient and safe. There are cryptographic voting systems in which the government publishes the entire result; everyone can see that their own vote (only) was counted appropriately; anyone can see that the overall vote was fairly counted; empowered auditors can randomly audit to ensure all votes are real. That is far superior to the existing, corruptible system.
Edit: adding some conditions on the crypto voting systems.
There are a couple points I don't think you've considered.
1. From my experience working in the lower class service industry (restaurant worker), Tuesday (federal elections) is usually a pretty easy day to get off. Weekends or government holidays that the middle class gets off, are the days where everyone has to work. My understanding is that this applies to retail as well.
2. You need non-technical people to trust the system. This isn't a technical problem, it's a political problem. Having one person from each party in the room while paper ballots are counted is actually a pretty good solution for this. Showing people who haven't done any advanced math pages and pages of whitepapers and equations that mathematically prove that the system can't be hacked doesn't really give you this. The system will be completely accurate, but rumors will fly and the government is quite likely to lose legitimacy anyway.
I'm a German that used to live in the US. In the US , voting sucks.
There is no "registering for voting" here. Ever citizen gets a piece of paper they bring to them to their assigned voting location(the room is assigned also, if you vote in person you have to go here). The voting places are always reasonably close. Like minutes by walking. You can opt to get a mail in ballot, if you want. Voting is always on Sunday when most things are closed. The vote turn out is over twice as in in Germany. The US has one of the lowest voter turn outs.
But our ballots are big and easy to read/mark. No stupid "dangling chad" crap like in FL in 2000.
Early voting is available over the weekends in many places. A weird amount of people don't realize it's a thing and believe they have to go on election day.
Making voting hard makes the system ripe for abuse, and that's exactly what we see happen all the time.
What we need is a system that is efficient and safe. There are cryptographic voting systems in which the government publishes the entire result; everyone can see that their own vote (only) was counted appropriately; anyone can see that the overall vote was fairly counted; empowered auditors can randomly audit to ensure all votes are real. That is far superior to the existing, corruptible system.
Edit: adding some conditions on the crypto voting systems.