I once asked this question of Dr. Woods in an online chat and he referred me to some material written by Walter Block which I haven’t had the chance to read yet, so you may want to search that out for yourself.
In any case, it is important to remember that in an AnCap society, your reputation would be incredibly valuable, and “community standards” would still exist, despite not being legally enforceable in a way that we recognize today. In other words, most people find child abuse abhorrent and intolerable behavior, and it is entirely possible (and I would say quite likely) that most individuals in a community would refuse to do business with someone who is known to brutally beat their children. Such a person would likely pay a hefty penalty for their behavior, not in the current sense of “people will grab you and lock you in a cage” but in a more subtle sense of “it will be harder for you to get a job” or “you might not be able to purchase goods and services you desire.”
Also, let’s not pretend that “it protects children” is somehow an inherent quality of government. Throughout human history there have been many far more restrictive/oppressive/comprehensive governments that have done no such thing. In some places in the world, this is still the case.