Paper voting is comprehensible by everyone.
>No single person can even fully observe a single counting station, much less a whole election.
First is possible (just observe the person whos counting), second not, for obvious reasons that you can't be at two places at once.
.. and trust the votes to all be real?
A person voting could put two votes instead of one. (It can still look like one, using the same method that's used for card tricks.) Even if you introduce extra steps, like every paper needing a stamp or something, you're still going to need to trust this stamper person. They could be in on it, and either stamp multiple or leak the stamp design.
The staff could be inserting fake votes into the ballot box before it gets counted, even without actual voters.
The counters could add votes from their sleeves, again classic card trick mechanics.
The counters can remove votes from the pile by stacking two or just sliding one off the table while also performing a distraction. Once again, classic magic trick mechanics that are used for vanishing jewelry etc.
Basically my point is that if the counting station is being staffed by magicians from Vegas, they can produce any result they want without an average person that's observing understanding anything.
Besides, at the end of the day it doesn't even matter, because one counting station is within the margin of error for the whole election.
I'm a Greek citizen. At the voting place of the city where I was born and am registered since, there are many rooms. Depending on the initial letters of your last name, you have to go and vote in a specific one. Each room has at least two randomly conscripted citizens, responsible to strike out your name from the list they have, once you drop your single vote envelope. Reminds me of the jury duty of USA citizens. Every few rooms have police officers (very possibly also randomly chosen) inspecting and protecting the procedure.
What really makes the paper ballots secure is that counting fraud does not scale. It is impossible to do it on substantial number of voting stations with nobody noticing even in semifunctional democracy. With electronic voting scaling an exploit to all voting machines is straightforward.
The Australian Ballot works pretty good. Private voting, public counting.
https://en.wikipedia.org/wiki/Secret_ballot
Like all systems, requires funding and attention, of course.
Comparatively...
There is no electronically mediated equivalent of the one-way hash of dropping your ballot into the box suitable for elections in the USA.
Because our ballots are complicated and our units of administration (precincts) are small. Meaning any end-to-end auditable system for hashing ballots won't generate the necessary hash collisions to hide one's ballot within the herd of ballots.