While #1 is generally true, it’s not true in all cases, or at least depends upon how the parents in question define “pressing interest.” Parental child abuse is hardly unknown, and the sort of parents who are prone to abuse their children will not pay a protection agency to enforce the child’s claims against themselves (and children will lack the resources and, being minors, the ability to contract with a protection agency themselves).
Your #2 ignores the original premise – even if the grandparents learn of child abuse, they are not themselves the victims. As OP wrote, under Rothbardian anarchism “ only victims of a crime could take the respective criminal to court” – grandparents aren’t the victims of the abuse of minor children, neither are third parties who might witness it, school teachers who notice bruises or other signs of abuse, doctors who see signs of child rape, and the like.
Look, I’m certainly not going to hold up today’s child protection as a model of excellence (it misses much real abuse, and often persecutes people who have committed no real abuse). But here is something where IMO OP has a point – it’s not even an instance where “well, given how bad things are under the current system, we should at least give AnarchoCapitalism a try before saying it would be worse” – it is not even theoretically better in this instance.*
*I focused on parental abuse because I do think an AnarchoCapitalist society could devise a means of allowing the parents to be a child’s proxy – while rejecting those who sometimes write as if the parents hold their children as a sort of property or homestead them somehow. I’m in agreement with Stephan Kinsella that this is an odious doctrine. But it would seem there is no way, consistent with the Rothbardian doctrine mentioned above, that the AnarchoCapitalist society could protect children from abuse by their parents. I’ve seen it written – for example Mr. Kinsella expressed this to me – that while the general rule would be parents would be entrusted to be the proxies of their children, if they were found to be abusing or neglecting them sufficiently, that prerogative could be reassigned to another. However, the gap in this remains that such a third party would not have standing to bring a suit to have the custody over the children reassigned from the parents to themselves, because that third party would not be the victim. Thus they could not, consistent with the Rothbardian principle that only a victim (or, extended, only the victim’s existing legal proxy) can bring the victimizer to court – they would be bringing a case to court to gain the standing they do not have, and the case would have to be dismissed. Or the Rothbardian principle violated, but to violate it is to effectively reject it.
This is one of a couple reasons why I’m not an AnarchoCapitalist (though I do want to give a hat tip to Stephan Kinsella for giving me some very thoughtful and considered replies when I contacted him on this issue, which made my more sympathetic to at least his version of how children would be treated & protected in a Rothbarian society. But I still found it problematic).