enough...you do know the two stories in Matt and Luke are incompatible and that they reflect just 2 of any number of Jesus birth traditions. The church Xmas nativity stories today in the Western Christian tradition are an attempted blending of the two. The Greek Orthodox tradition however maintains Jesus was born in a cave. In this they agree with Justin Martyr and Origen who both said a cave in Bethlehem was the site of Jesus' birth. The Protoevangelium also testifies to this early tradition. Emperor Hadrian even went and leveled the site because first century Christians were making pilgrimages to the cave. Ascension of Is has him born in a house like Matt..
Then we have Ignatius, the supposed first attestation of Matt, he has a very different version of the star story where Jesus IS the star.
Now the virginity of Mary was hidden from the Prince of this world, as was also her offspring, and the death of the Lord; three mysteries of renown, which were wrought in silence by God. How, then, was He manifested to the world? A star shone forth in heaven above all the other stars, the light of which was inexpressible, while its novelty struck men with astonishment. And all the rest of the stars, with the sun and moon, formed a chorus to this star, and its light was exceedingly great above them all. And there was agitation felt as to whence this new spectacle came, so unlike everything else above. Hence every kind of magic was destroyed, and every bond of wickedness disappeared, ignorance was removed, and the old kingdom abolished, when God appeared in human form [or revealed himself in a human manner] for the renewal of eternal life.
Ignatius, Letter to the Ephesians 19
We haven't even started on this topic, but my point is that there simply wasn't the a singular tradition about Jesus' birth in the earliest Christian records. No one appears to be fixated or maybe even aware of the contradictory birth narratives in Matt and Luke.