Ok that's not a convincing exposition of the text. It says he was conceived in sin.
Yes, IT says he was conceived in sin because David himself wrote that. That's no doubt how he felt at the time; he was human, and he was mortified that a) he had sinned against God and b) he had been found out.
That doesn't mean his words are a doctrinal statement that all babies are conceived in sin. You haven't explained Psalm 139 - why does he say "I praise you because I am fearfully and wonderfully made; your works are wonderful, I know that full well." Psalms 139:14. Did he contradict himself, or was he saying that babies are wonderfully made by a God who does wonderful things - yet they are made in sin?
The Bible was not dictated from heaven to a robot; it was written by human beings.
Yes, of course they were guided by the Holy Spirit, but they were human, and sometimes the authors' joys, sorrows and frustrations come through. Read some of the Psalms - David wrote that he was lying on a bed of tears, that his bones were weak, that life was draining from him. Some of the prophets were sad, and angry, at the sin of the people and that they were abandoning God.
Paul was furious with the Galatians for abandoning the Gospel and following another one. He wrote "I wish those agitators would go and emasculate themselves!" Galatians 5:12. He was very frustrated with those who were teaching that circumcision was important for salvation. So the text says that; it is Scriptural. But I doubt anyone believes that the Bible is teaching that all men need to have some drastic surgery.
it would seem to apply to all babies.
No, there's nothing to say that that is a doctrinal statement that is true of everyone.
God made all people in his image, Genesis 1:27. There is nothing in the world that was not made through Jesus, John 1:3. We are told in the Psalms that children are a reward/gift from God. We are also told in 1 John 4:8 that God IS love.
It is not an act of love to create a baby, and then declare that that baby is sinful and will go to hell for rejecting Jesus, when it wasn't the baby's fault that it died before it even met Jesus.
If an earthly father tried to do that, or disowned the baby and wanted nothing to do with it; we'd say he was cruel and deserved to be charged with neglect. Why would we believe that God is like a sinful, human father?
The character of God that I see portrayed in Scripture tells me that he would not end babies to hell.