I doubt that a professional translator would propose “they do nothing (at all)”, because that's definitely not the meaning of the sentence.
The adverb “anzitempo” (which can also be written “anzi tempo”) comes from the Latin ante tempus, that is “before the (right) time”. Hence
They do nothing too early
which might be a way of saying that they do something only when it's the right time to.
The Latin ante gave two Italian words: anti and anzi. There is also anti from Greek, but with a different meaning.’
The former is used only in composites such as antipasto, anticamera; it can be ante when joining a verb, such as anteporre. It means before in space or time.
The latter is the main object of the question: anzi can be a preposition or an adverb, but the usage as a preposition is uncommon in modern Italian, except for anzitempo, anzitutto and some other forms that are literary anyway; you can see examples in the linked entries of the Treccani dictionary. As an “independent” word, anzi became also an adversative adverb and this is the most common meaning in modern Italian.