I disagree with your depiction of God using words like extermination, as if God is evil and His purposes are not good. From my point of view it lowers the accountability of human beings at that time. I believe they were accountable for their sins and yes it even affected their children, who were killed along with their parents in a few cases. If God were true, then it makes sense to me that He has every right to take away the lives that He gave in the first place. For His purposes. It may be entirely possible that those children were given eternal life.
What word would you like to replace "extermination" with? How are you going to spin such a word around so not to emit negative connotations here? People die. People die at the 'apparent' order of God. People die at the 'apparent' hand of God. Civilisations have been eradicated at the 'apparent' hand /order of God. If you replaced God with any dictator throughout human history you'd be cursing them, but because it's God who has done it then somehow you flip 180 and it's all fine and dandy.
....And what about free will? What happened to the free will of these children? I keep hearing that God has free will on pretty high priority, if not the highest, yet here you have the purposes of God superseding it. What have these children done so that they deserve to be slaughtered? If you can't think of a valid reason why innocent children deserve to be killed, and lay it all on "well God knows why and knows best" kinda bullshit, then you have surrendered your compassion and integrity to sycophantically follow a dictator who you find unquestionable.
This is where this moral quiz that apologists throw out gets flipped on its head, the "would you torture a child to death just for fun" nonsense. Well no, I wouldn't, and I wouldn't torture a child to death just for God either. I can't say I'm too confident that your answer would be the same.
Oh, and while in this theology it may be possible for these children to be given eternal life it's also possible that they won't, and since you don't consider yourself the judge of who's in who's out, then just throwing the possibility out there to cover you and your Gods arse serves no purpose but to show that you will do anything to spin the slaughter of children into a positive. It's disgusting. No, it's fucking disgusting. You should be ashamed of yourself.