too-strong fear of a 
   lying false positive

You might be a bit optimistic here: it's not always easy to deal with "lying false positives". Depending on the legislation it may be difficult to fire somebody, and involve going to court, sometimes going on over several years. Vindictive personalities may engage in sabotaging the company, the team, their (former) co-workers, as a retaliation for being fired. (Anecdote: I have witnessed all of the above in my work.)

Summary: the cost of a false positive almost always outweighs the cost of a false negative.