Why could the birth narratives according to Luke be seen as historically inaccurate?

His chronology of Jesus' birth appears to be mistaken. He writes that it happened when Quirinius was the governor of Syria - we know Quirinius governed Syria, which included Judea, from 6CE to 12 C, but it can't be proved that he was governor 'in the days of Herod', who died in 4CE. There is no evidence of an enrolment or census before 6/7 CE. It has also been claimed the three hymns in Luke's Gospel were in use before he wrote his Gospel, so Luke is wrong to ascribe them to Mary, Zechariah and Simeon.