There's a limit to what a security background check can discover. Crimes such as soliciting sexual contact with minors over the internet are virtually impossible to detect because of the private nature of the crime itself. In other words, sting operations seem to be the only sure-fire way to catch people in the act. And when you think about it, this shouldn't be much of a surprise.
The violator usually cruises the internet from the privacy of his own home over a private internet service provider and under the anonymity of a screen name. Law enforcement agencies essentially go to known "pick up points" in the form of websites and chat rooms to wait to be approached. From that point, it's a matter of developing probable cause to obtain what amounts to an online search warrant to discover the true identity of an online solicitor.
To suggest that a background check should be able to detect these types of criminals is to suggest that government background investigators be empowered to cruise the internet on what amounts to fishing expeditions. This is a greater evil because it encourages very intrusive government actions all in the name of "security." Gotta draw the line somewhere.
The imperfection of current background checks is that they assume that if you don't have a criminal history, you probably can be trusted. For about 95% of the population, this is probably a valid assumption. However, there's that small percentage of people who can successfully clear a background check but who will nonetheless commit serious crimes.
One additional comment that has nothing to do with airport security screening: in my old line of work, it seems that the Army was looking for demons with saintly character. In other words, the type of work we were expected to perform requires an ability to divorce oneself from what is generally regarded as right over wrong. Intelligence collection is very dirty business; the people who excel at it have to be able to step outside the box without getting tangled up in the messy confusion that often results. It's difficult to explain without digressing on a lengthy tangent. Suffice it to say that the irony was that we were expected to be both devious demons with the backgrounds of holy saints.
Picture the confusion on the polygraph examiner when he asks the obligatory test question: "have you ever stolen anything in your life?" and the response is "f**king A' I have."