But that's only if one slavishly interprets the verse with the Animals outside and does not consider that Israel is a warmer country. It is mainly predicated on that verse. You can elaborate if you wish to, I will read it.
Herod dying in 4BC has been strongly disputed recently. The Wikipedia page reflects this:https://en.m.wikipedia.org/wiki/Herod_the_Great
It lists 4BCE(4BC) and 1CE(1AD) as two possible death dates, with alot of scholars now supporting the traditional date again instead of the 4BCE date. So that is in fact not known. We thought it was for a while clear, but newer archeological developments, such as the discovery of his possible tomb threw it back into doubt.
And of cause it wasn't an ordinary star, it was an extraordinary miraculous event. Even if we figure out it was Jupiter(the King star in most ancient beliefs), the planet had appeared to perform extraordinary movements and actions anyway, like appear to hover over Bethlehem and looked miraculously much brighter than normal. The Planets were given the name Planet because it means a "Star that wanders" which is a description which also obviously applied to the Star Of Bethlehem. It literally wandered around the night sky giving the "sign a king was to be born in Israel" then appeared to stay still and "rest" over Bethlehem with it's light appearing in the sky above the stable. The Magi traveled directly from Jerusalem so the point that if they had of "gone off centre the star would be off course" is irrelevant given they probably traveled as straight as possible for the "star" and found the stable directly in that direction which still means Scripture makes sense in saying the Star pointed out where they needed to go and was shining on it. They were traveling directly from Jerusalem and when they got to Bethlehem and walked up to the stable(directly under) they would have known that's where they needed to stop as soon as they saw a newborn child in a stable under the star. It is erranous to assume they did not know Bethlehem was where the Messiah was to be born given it was mentioned in their presence in Herod's chamber(and it can be assumed they knew the Jewish beliefs a bit), and therefore knew not to leave the small city once they got there.
The Magi were astrologically minded(Magi traditionally followed the Religion of Zoroastrianism, which placed naturally heavy emphasis on Astrology, although the Bible does not explicitly mention they were Magi of this type the name does imply they were), they didn't just see the star like everyone else. The Bible explicitly states they were in fact the only ones to see it, Herod didn't even notice it, no one in Jerusalem had apparently noticed it. That's why Herod interrogates them to see when they saw the star. They saw a miraculous sign in the sky that no one else saw and that no one else appeared capable of seeing or willing to see. Which supports the contention that the sky appeared normal to most observers prior to the Magi arriving in Jerusalem, but these individuals were able to read something miraculous happening that others just didn't see. Which supports the contention they were masterful Astrologers, able to read the stars properly(not selfishly sinfully like most modern astrologers do) as God had intended to happen when he created the stars..
Genesis point out God made the stars to help us read the signs and know the seasons. I'm sure you know that verse already.