Voting is not a new event. We've been doing it for years. If you were to research modern elections where machine tabulation was used then hand recounts were used and the results compared, the machines are more consistent.
And as to: " When I feed my ballot into the tabulating machine used here, I have no way of knowing whether my vote was counted at all, much less whether it was counted in the same way as I cast it. " To be honest if you slipped your ballot into a ballot box you have no way of knowing whether your vote was counted at all, much less whether it was counted in the same way as you cast it.
WW