According to the Bible, why is everyone born a sinner?

Answer:

As in Adam all die..

As sin entered into the world by one man…

The rest of those scriptures is the answer to your question. The central fundamental doctrine of Christianity—the Ransom.
Because Adam contaminated his entire race because of disobedience, it only takes one to ransom him and his entire race.
Israel at to make atonement with literal bulls and goats. Paul tells us that those sacrifices did nothing actually, but were a schoolmaster to point to Christ—who actually could do what the bulls and goats could not.

This is also pictured in the clothing that God provided Adam and Eve when they left the garden.
“Without the shedding of blood there is no remission.”

Jesus gave his life a “ransom for all” to offset Adam’s sin. The word “ransom” means “corresponding price”. Just as in our day when someone is kidnapped, they pay a ransom—what the kidnappers decide is a “corresponding price” for the one they took. A perfect human life for a perfect human life is the corresponding price (not an animal or angel or god—but an equivalent for Adam before he sinned)
​
The application of the ransom is a different matter—that is why the Bible teaches a Resurrection and a Kingdom. But before those happen there is a once in the universe calling for the Bride of Christ—that is what we are in now. We are called to be kings and priests—in that Kingdom we pray for—to uplift all of mankind when they come back in the resurrection—paid for by Jesus.