> It is certainly the case that trust in such machines could be misplaced.

Right but if that's the case, you're going to have to count them by hand at some point. So what do the machines bring to the table, if their whole purpose is to avoid that?

You usually only need to hand count a small number of randomly selected votes to verify the result statistically. The exception being very close results where a handful of votes makes the difference.

That is a good point, as long as "randomly selected" can be assured.