Mandatory random audits are part of the system. We get automated counts from the Scantron machines, but those are unofficial, preliminary. On election night, there are random audits of some number of voting machines for hand-counts, to detect systematic fraud. And the total number of ballots in each machine is hand-counted to make sure it matches the totals given by the machine.
Paper ballots that are hand counted. Ideally, streamed live.
More seriously, France has paper voting, yet results are published at 8:00:00pm, closing time of the last voting offices. How? Statistics. And only takes the night to confirm, when we’re on a 50,3% kind of results. And it’s already 60m voters; If it works for 60m people, it will work for 300m (especially if only 120m vote in USA).
Or you publish all the marked ballots, so that anyone who wanted to could do their own count to verify the results.
You can also design this so that any individual voter can check to see that their ballot was included in the count, and counted correctly, without being able to prove to someone else that they voted for a particular candidate (so that vote buying schemes aren't enforceable).
See Scantegrity [1] for a specific system with these properties, which can be implemented on top of existing optical scan vote counting machines. For a more general look at securing voting, see end-to-end auditable voting systems [2].
[1] https://en.wikipedia.org/wiki/Scantegrity
[2] https://en.wikipedia.org/wiki/End-to-end_auditable_voting_sy...
To use a metaphor, it's not having 2 locks on your front door, it's having 2 different keys that open the same lock.
Say you have a hybrid system, and the numbers come in and the electronic and paper counts disagree. Which one is used as the true vote?
If you take the paper, then what was the point of the electronic? If you take the electronic what was the point of the paper? If you throw out all votes where the paper and electronic disagree, you just enabled any "fraudster" the ability to disenfranchise any subset of votes by breaking either system.
You also can't exactly pick and choose which to accept per instance (could you imagine the abuse that could come from that!?), and methods of "averaging" them don't really solve the problem as much as they make it marginally harder to pull off (to sway 1% of votes in a 50%/50% weighted system, you'd only need to hack 2% of the electronic machines). And of course using a significant "disagreement" between paper and electronic to trigger re-doing an election isn't a solution either because if they hacked it the first time, chances are they can do it again. Not mention that having a second voting day because the first was taken over by fraud is going to be a logistical nightmare, it will change who can come out to vote, and it may even change who/what people will vote for.
If what you want is an audit or a second count, then call for more exit polls or an explicitly secondary system. But this whole dual-voting idea isn't really a good one when you really dive into the dirty details of how it would work.
If the result is 49.999% vs. 50.001%, you cannot conclude anything meaningful about the will of the people. Maybe the weather was just slightly worse in a neighborhood supporting the losing candidate and it tipped the result slightly.
So even if the voting process is perfect, that close of a result is totally random for other reasons.
Most people don't seem to be capable of thinking about non-determinism like this, though. Witness the huge number of Americans who think the presidential election is illegitimate because Trump got 46% of the vote to Clinton's 48%, when honestly 46 vs. 48 is very little difference in terms of what they mean about the general will about Americans.
On the other end of the political spectrum, witness the amount of Brits who think a 52-48 vote to leave the EU is somehow relevant.
In calculus/analysis terms: this is why proportional representation parliamentary systems are better -- because the function from "how many people support which parties" to "what policies you end up with" is not discontinuous.
If the UK ran on a proportional system, they'd be getting some sort of very soft, Norway model Brexit. If the US did, a center-right Democrat like Clinton or Biden would be PM.
Routine double counting all votes would be controversial due to the high cost.
But you'd need mandatory random audits, and a legal requirement to keep the paper originals for up to years.
There's been cases of the paper ballots being destroyed while legal action was ongoing, and a never audited system cannot be trusted (because there's implied trust, not tested trust).