Re: The self-created problem, easily solved
I think a lot of people outside the US don't understand how our elections are conducted, and why we can't use paper ballots and manually count them that evening. Elections are conducted by the states, and assigned to the counties, which divides them into precincts. You don't get a ballot for president with a few choices, you get a ballot for president, a congressman and then a whole host of state, county and city positions ranging from governor (though most states do those in years like 2018 where there aren't presidential elections) to state water commissioner, election/approval of judges in some states, county board of supervisors, auditor, treasurer and so forth, city council, probably even animal control officers in some places. Then you might have a number of measures on the state and local levels to approve or disapprove.
Another problem is the size of precincts, which tend to be pretty large in the cities. To no one's surprise, the largest ones are in predominantly black areas in big cities (especially in swing states like Florida) because it acts as another way to suppress their vote by making them wait in long lines to vote. Where I live I have never had to wait in line for more than two minutes, and haven't ever seen or heard of lines that extend outside the polling place where bad weather will also help suppress the vote in overly large precincts.
The ballot where I am is typically a 10x17" sheet of paper, with two columns of items on both sides. You really think people can accurately count all that stuff by hand at night after they've been volunteering at the precinct since probably 7am or so, especially in the really large precincts? This is the "paper ballot" I get, which as far as I know is pretty standard for the US, which I feed into one of those bubble scanner machines to electronically score. Now sure, some of would suggest the US could have a different election day for president/congress and make hand counting easier, but I don't think that's necessary.
Even if you want to use fully automated touch screen machines to vote, so long as it produces a paper trail and the voter is encouraged to check it before putting it into the ballot box then you have no problem. Electronic distribution and counting of votes so you can quickly announce the results as demanded by the media is fine, provided you later verify that electronic tally by inviting representatives of political parties to participate in a mandatory recount the next day.
They count things up, and send their totals in to be posted on a website with all the other totals nationally, and automatically added up and shown alongside the electronic tallies. As with Stork's Denmark example, those present at the precinct could check the web site to make sure the recount tallies reported are what they recounted.
If there's a discrepancy of a few votes here and there that don't change the result, no worries, but obviously you'd have processes to deal with close results where maybe the discrepancy matters as well as precincts where the discrepancy is large and something obviously went wrong somewhere.
Personally I would only require manual recount of a few percent of randomly selected precincts (the sample size and procedure for randomly selecting precincts determined by a panel of statisticians) and so long as they are all within a reasonable statistical margin, you consider it good (unless there's a challenge that requires a full recount, but the party asking for the recount should pay for it unless the result is within some small margin statisticians will also determine) If the few percent count isn't within a reasonable margin then you conduct a full recount automatically paid for by the state, and any statistically significant deviations are thoroughly investigated and remedied.