Another explanation about "death of the son" - that you decree upon your son and purify us from his sins. This refers to the Israelites, who are called children of God. Another explanation about "death of the son" is about the sins that the son commits and the Day of Atonement comes and atones for him and forgives him. Another explanation about "death of the son" - the rabbis said that as long as that son was alive, his heart was like the heart of the Almighty upon him. But once he dies, it is known that his heart is not upon him anymore.
What 'Death of the Son' Really Means as Purification
Midrash Tehillim 9:3