"Think of the children" doesn't rationalize even one wrongly accused person - especially in today's society where an accusation is enough to completely ruin ones life.
A false negative could ruin the child's life just as easily, so I suspect most parents might think that's more important. It's not necessary to choose either "privacy uber alles" or "think of the children". As with most things in life, there's likely a good balance somewhere between the extremes.
For example? A child being preyed on by a pedophile? What would stop that from happening in real life? Maybe we should set up cameras every five feet since obviously if we could stop it (no matter how immoral we have to act to do so) we have an obligation to, right?
People have obviously found that online fora are perfect to groom children, because you can approach a child far easier than doing so in a playground. You can also repeatedly try with different children without your likeness being plastered on a wall somewhere. Since it's an obvious target, and checking it can be automated, why not police it in a balanced way? Again, you only see one side of this.
Similarly, in real life if you had a cheap, automated way of highlighting interactions that have been proved to have a higher chance of leading to children being molested while still balancing that with the human rights of the adult, most would find a balance between "do nothing" and "be hysterical". We don't have such an automated mechanism, and I'm sure your cameras-every-five-metres idea is on the hysterical side of balance, so even using it is obviously a straw man argument. That doesn't discount more rational approaches for those who have more than one principle in mind.
And be serious. Getting asked a question from Facebook is hardly going to ruin someone's life is it now ?
in case of law enforcement, the only acceptable false positive rate is zero.
Laughable. It does when Facebook are going out of their way to read something that is absolutely none of their business. You can easily obtain zero false positives by not turning anything over the the police.