Sampling iid would be unconstitutional and also obviously politically infeasible. So, the only people who are investigated are people for whom there is probable cause or at least an accusation.
Once you're sampling from a population for whom you have either probable cause or at least an accusation, the group under suspicion probably not an iid sample of the overall population and may even be skewed in the wrong direction. Eg, a physical abuser is more likely to conceal a bruise than the parent of a particularly careless child.
Additionally, suppose for sake of argument that abusers are in fact over-represented in the accused group. The behavior of both abusers and the abused confounds the statistical analysis. Consider alcoholism as an analogy (but also very correlated underlying pathology). Just like with alocholism, there are "high functioning" abusers who are able to avert suspicion and, in case of suspicion, manipulate the people who they are abusing.
So we are sampling from a constrained population in which abusers may in fact be under-represented, and then again within that group there are confounding issues.
None of this is to say that we should abandon constitutional rights, of course. But in a country with strong protections for the accused we might actually expect certain types of crimes to be more difficult to investigate and prosecute than a "just iid sample" approach.
(It's also possible that the estimates are wrong, and in both directions. It's additionally possible that the 16% statistic refers to a different definition of abuse than the 5% number. Etc. etc.)
Yes a kid who ran into a doorknob is more likely to show up at school with a visible black eye than the kid whose mom is used to covering black eyes with makeup, but CPS knows, or at least should know this and alter their behavior to spend less time investigating a single unconcealed black eye and spend more time looking at the kid wearing concealer on one eye on a regular basis. The whole point of having trained professionals investigate is because we know that accusations are a noisy signal that must be filtered.
I fully agree that the 16% number should not be treated as a target. For starters, that's just sexual abuse, the rate of physical or sexual abuse is necessarily higher. But the point is that when evaluating a system you can't simply say you find abuse at a non-zero rate and thus it's working - a broken clock is right twice a day, and doing anything at all will find something. You have to compare to a placebo to see what effect your actions are having. If you're substantially worse than the placebo, that doesn't mean sugar pills are the cure, but it means you need to do things differently.
I would love to see evidence there, I think that's wishful thinking. Requiring warrants will simply mean fewer searches. I understand those who think that's a fair balance for individual rights. But I don't see it uncovering more abuse. I see it uncovering less, resources being kept equal.
Who is estimating 16% of children are abused?
It doesn't sound correct at all.
On it's face the simplest explanation is child protective services is investigating a different definition of "abuse" than whomever claimed 16% of children are abused.
Estimates are that around 16% of children are abused, so if you're only finding 5% it means you're substantially worse at choosing households to investigate than if you were picking at random. Of course that's a national average and some places may be better or worse, but really all that indicates is that resources are being diverted to the wrong areas. Every hour of time and every dollar spent investigating households without good reason is stolen from a victim who truly needed it.
Obviously not every investigation needs to find abuse to be justified, the whole point of the investigation is we don't know if something is wrong or not. But just because there is an occasional success does not mean that the current system is working, nonetheless optimal.