This is absolute hogwash, there are other methods than a full hand recount if you have a paper trail, some of which only require counting a small number of the ballots by hand.

The best example of this is a Risk Limiting Audit (RLA). You only have to re-count a smaller number of ballots until the overwhelming probability is that the vote is confirmed, or that the vote is rejected. Depending on the disparity between the ballot options, this count can actually be very small.

See: https://www.stat.berkeley.edu/~stark/Preprints/gentle12.pdf

This system is perfect for this kind of an audit -- essentially a ballot marking device written by an organization known for formal verification.