Why don't the Four Gospels record the exact date of Jesus' birth?

[Root of Jesse]

Why don't the Four Gospels record the exact date of Jesus' birth?
As we know,
1. Jesus was born in 4~6 BC, not 1st AD.
2. He was not born in winter because the shepherds were taking care of their sheep on the plain outdoors. (Luke 2: 8 And there were shepherds living out in the fields nearby, keeping watch over their flocks at night.)

Did the authors know exactly Jesus' birth date?
Why not they asked his mother Mary?

Because even Jesus' closest colleagues only knew him as an adult. Only John took
Mary into his care, and birth dates weren't as important to them as to us.

But there is really no reason to believe that Jesus wasn't born in winter. Astronomically, there is reason to believe that He was born exactly when the Church says it happened.

 

[Root of Jesse]

It was too cold? What does caring for sheep involve?

Remember...sheep have woolen coats.