Rushing is faster, unless rushing causes mistakes. This is why people rush. And they usually rush during tasks where mistakes are not likely and/or mistakes are mostly inconsequential in the event they happen.

Slowing down can feel better—it can induce less stress. And it can help you avoid time-wasting mistakes, as you've mentioned.

There are plenty of exceptions where a little bit of "slow" time spent planning before engaging in a hurried task can reduce the total time spent. This might be a good compromise.

But as far as overall net time spent doing menial tasks, I'd guess rushing buys you time...unless you are a complete klutz.