Hand counts is the way it is done in Sweden. It means that anyone can easily (well, as easy as possible) both understand the system and inspect it locally (all counting is open to the public).

The result is delivered in a matter of hours, and since vote counting is a parallell process it scales well enough that I doubt it would take much longer in the US. The US have massively more complicated ballots though, which I think is another issue entirely that you guys must solve.

I doubt it would be very expensive either. You don't have to pay people very much to do their civic duty every couple of years and count a couple hundred ballots.