Yes you are 100% correct and that is why some Christian denominations don't celebrate Christmas. But no Christian fully follows The Bible and just picks to follow what they want to accept. According to the bible they should drink poison as a test of fatih (which should not hurt them according to the bible). 99.9% of them do not do this even though the bible tells them to do so.
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The Bible does tell us of only two birthday celebrations. One in the Old Testament (Genesis 40:20-23), and one in the New Testament (Matthew 14:6-10). In each circumstance, someone died as a result of wishes of the person whose birthday it was. There are no favorable birthday celebrations mentioned in the Bible. There is also no record of Jesus celebrating his own birthday.