Babies are born into a sinful world, yes. But they are not born in sin.
If they were, then firstly, Jesus would have been too because he was a human baby; God, yes, but still human. Secondly, any babies who died would go to hell. Even if they only lived for 10 minutes, they would have died before they were able to hear the Gospel and be born again. And thirdly it would mean that God was allowing them to die before they could learn they were sinners, hear about his amazing gift, Romans 6:23, and receive Christ. Then he would be punishing them for not having lived long enough.
Sounds like you are struggling with a issue many people have trouble with. Not just infants but everyone who has never heard the gospel. You are offering a solution. I believe the scripture says we were sinful from birth. The question is are they culpable for their sin ;seeing they cannot do anything about it themselves and the actual solution has never been revealed to them? I have struggled with culpability for many years.
How can a blind, deaf, and dumb person be held culpable for a fate they are helpless to avoid? The old Who song "Christmas" speaks of Tommy, the blind deaf and dumb boy:
"
Did you ever see the faces
Of the children? They get so excited
Waking up on Christmas morning
Hours before the winter sun's ignited
They believe in dreams and all they mean
Including Heaven's generosity
Peeping 'round the door to see
What parcels are for free in curiosity
And Tommy doesn't know what day it is
He doesn't know who Jesus was
Or what praying is
How can he be saved from the eternal grave?"
There is a clue in the words of Jesus: “Let the little children come to me and do not hinder them, for to such belongs the kingdom of heaven.”
Others suggest that those who have not heard the gospel will be given a chance in the afterlife. Others suggest that God knows who, amongst those who have not heard, would have believed if had they heard.
I can only believe that God is ultimately fair, just, and merciful. The idea that our God would cast billions of souls into an eternal burning torment is just too much to conceive.