What is written in this verse is a beautiful witness of Amaziah. In the first place, because he does not do what was the usual procedure for kings in that time, namely killing the whole family. With the brother nation Israel this did happen (see 2 Kings 10:1–17). Against the culture of the day, Amaziah acts according to God’s commandment in Deuteronomy 24:16 (the quote in 2 Kings 14:6 is almost identical to Deuteronomy 24:16).
That is a message that a preacher of God’s Word must always bring, under all circumstances: our thinking and acting as Christians is not determined by the surrounding culture, but the Word of God, especially when it goes against the culture. But this is especially a beautiful witness, as Amaziah in his actions shows us something of what it is to be a Messianic or a theocratic king. It is not about securing your own throne as well as possible. (In that case, he should have killed the children as well, because they were also potentially dangerous due to their feelings of revenge. And yet, that is not the cause that later on in life he was murdered himself, see 2 Chronicles 25:27).
Amaziah prefers reigning with a possible attempt on his life rather than shedding innocent blood; and this is how the king of God’s people ought to be. It is about justice, not about power—to minimize the violence as much as possible, and to strive for peace. This is how God is, and so is Jesus Christ. And something of this shows in the actions of Amaziah in this verse. Let's remember that Jesus, when he revealed himself, said and showed that his kingship had a completely different character (see e.g., John 18:36: My kingdom is not from the world
). The great Son of David moves all the other offspring of David into the shadow, though they could sometimes show a little of how God meant his kingship to be with his people.
6 But he did not put to death the children of the murderers, according to what is written in the Book of the Law of Moses, where the LORD commanded, “Fathers shall not be put to death because of their children, nor shall children be put to death because of their fathers. But each one shall die for his own sin.”