There is a minor holy day in ancient Israel called “The Day of Implantation” when the High Priest would send out his representative to sow a virgin field with the seed of the First Fruits offering of the barley harvest on the Feast of First Fruits on the Sabbath that followed the Passover. According to the Talmud (Mishnah, Menach. Viii. 1,2), exactly 70 days before the Passover, two priests would be sent forth to sow the seed for the harvest in a field that hasn’t been plowed or dunged (Mishnah, Menahot viii. 2), and one priest would do the actual sowing while the other priest stood as an observer and to announce to the field (and those who would follow the priests out) that it has been chosen as a holy field set apart to God.
If Jesus were born on the first day of Tabernacles in 2 BC…and Mary had a perfect pregnancy, this would be the very day that Mary was impregnated with the seed of God. One Priest, the Holy Spirit, came to sow the seed in a virgin field that had never been plowed. Another priest, Gabriel, came to announce to Mary that she had been chosen as a holy vessel to bring forth the Messiah! Then, 267 days later (which is the perfect time of human gestation), Mary gives birth to Jesus who is the First Fruits born from spiritual death (I Corinthians 15:20,23; Romans 8:29; Colossians 1:18) on the First day of Sukkot (Tabernacles). SOURCE
That said, if we are to celebrate Christ’s birth at all then our Heavenly Father would not hide His incarnation date from us but reveal it in His Word. He would have it plainly spelled out for us. For the same reason Moses’ body could not be found for fear of it becoming an idol to the Israelites so God obscures Christ’s date of incarnation so that the infant Christ is not idolized and worshiped. As we can see, the apostate religion of the Scarlet Harlot and her daughters have found a way to do just that on a day not even close to the estimated time on the Jewish calendar. The world does love the harmless baby Jesus and many supposed Christians join with them whether in ignorance or outright will worship.