Every time I try to get to the bottom of this, it always boils down to "trust the system" which makes me uneasy.
IMO the best solution here is to have electronic counting with an auditable and traceable paper trail as a backup. Every time I've voted for the past 10 years has been like this. First, I get a ballot paper from the front desk and stick it into an airgapped ballot marking machine. I then make my choices and the machine prints them onto the ballot paper. I'm able to read the paper and verify that it matches the choices I made. I then stick it into a separate airgapped ballot counting machine, which scans my ballot and deposits the paper copy into a sealed box. The entire process of setting up the machines, transporting the paper ballots, and reading the results from the machines is cross-checked and signed off on by volunteer poll workers from both parties.
> how can a constituent know with absolute certainty that their vote was counted
The representative of your party plus independent observer said all votes at your polling station were counted. You know both those community members and know them to be generally honorable. Ergo your vote was counted.
> every voter in the system was legal
None of the observers at the polling station, or the station head claimed any illegal person voted.
> the final tally was authentic
The observers all signed as witnesses on the final tally.
This is not the "system. it is humans you know who are telling you what they saw. If you can't trust other humans at their word, democracy cannot fundamentally work.
This, but also, important to point out that this is a question of scale: "If you can't trust other human*s*" - plural.
Surely we could do better? Testimony doesn't assuage my concerns that the process may not be tamper proof.
You should trust political volunteers after you have seen their track record of being honest and truthful. (Though there is some default amount of trust the process gets because of the adversarial nature of volunteers with opposing biases checking the process).
This is along the same vein as
You should trust candidates for the seat after you have done your due diligence that they have honest and truthful, and will faithfully represent you in the legislature/administration.
as well as
You should trust civil servants to have done state activities justly and produced truthful records and reports of state activities after you have seen a record of them doing these things correctly over time.
Democracy with humans is built on a lot of trust in humans. We have to keep this in mind when arguing about these things.
You do not have to watch every district, every election, every time. But given that enough people do it, at least once, at least in their own district, then it is easy to see why the system as a whole is trustworthy.
It's not that developing voting software should be open-source, its that actual voting should be "open-source" in the physical sense.
Trusting the system is possible if you can (you, yourself) readily observe every part of the system. I don't think giving members of the public access to the server your voting software is hosted on is a very viable idea, but giving members of the public access to paper count centres is (it's done very successfully in many countries).