They are faster and more economical than hand count, and preserve the option of a hand count to verify machine-count results. With careful work, they can probably be made as accurate as hand-counted ballots. Carefully-designed procedures for random recounts by hand can probably bring the likelihood of fraud in machine ballot counting down to acceptable levels, at least for optical ballots. Thirty-six hours is about the longest a person can stay continuously alert. If an election continues for more than 36 hours, either the operation must be stopped then restarted, or all responsibilities must be handed over from one person to the next.
Errors thrive in such moments of confusion.
Naturally, where there is potential for error, often there is potential for fraud. The ideal vote-counting system is swift as well as accurate. But while we are waiting for that better system, it would be better to use hand counting in places that currently use punch-card or DRE systems. Mechanical voting machines have in inherent weakness, in that each individual's ballot is not recorded, only totals of votes for all people who used the machine, so a meaningful recount down to the level of the individual ballot is not possible. This problem makes mechanical lever machines undesirable.