It's not "the" answer, but it has some serious benefits.
First, if you don't register your children (anywhere...) then they aren't in the system and easy find.
Second, by not registering your children you are taking away the presumption of ownership by the state/feds and so any pretense that they aren't actually kidnapping your children (which is a crime).
While it is true that most people don't understand that registering something in effect gives control to the ones you register with, lack of comprehension does not change the facts.
And since, as you note, they believe they can pretty much get away with anything... and most of us don't know enough law to file a lawsuit properly much less pursue it... then they can and will still do these things.
Not so many years back (15ish?) a lady down in oregon that was prosecuting local officials for child abduction and pedophilia had her kids taken by CPS... and the one that wasn't registered (the youngest) was returned the next day.
Of course, add bolder criminal elements in official places to average people who have learned nothing about how law works and you end up with serious abuse.
I agree. It is however very hard not to register a child, depending on where you are. My youngest 2 were born unassisted at home so I could have gotten away with it easier. But with my eldest son, my then 'lovely' parents in law made an anonymous referral to the child protection service in Holland. So even though he wasn't registered at the time, they still got involved even though he was perfectly fine and it is not illegal to have an unassisted birth there. The unassisted birth of my youngest is in our file now in this case here in Ireland. Since most people travel and need passports, it is needed for those to get their children registered. It's not fair and I personally don't believe in man made borders, but that's the way it is and there is no running from it.