OK, here we go . . . here’s our 2 cents worth on when Jesus – Yeshua- was born. By the way, just as a caveat, if this information and understanding was relevant and crucial to our relationship with Father God, it would be laid out in such a way to reflect that relevance and necessity. Now, with that said, let’s begin . . .
The Word was given at Sinai (see Exodus 20 and following) which we understand to be the first Pentecost or Feast of Weeks. John declares that ‘the Word was made flesh and dwelt among us . . .’
Our premise is this – if the Word was given the first time at Pentecost, then it would seem to follow suit that the Word became flesh ( Jesus’ birth ) at Pentecost as well. Ok, so let’s see if it fits.
Zechariah, the father of John the Immerser, was of the ‘order of Abijah’. 1st Chronicles 24:10 shows us the order of Abijah was the 8th order. Each order served for 2 weeks ( or half a month ) therefore Zechariah’s weeks would have been weeks 15 & 16 ( the last half of the fourth month of the service order).
2nd Chronicles 7:5-9 tells us the dedication was at the time of Sukkot (Feast of Tabernacles) . If that’s the case, then weeks 15 & 16 of the priestly order of service would fall somewhere around the time of the end of February or the first of March on the modern calendar. IF . . . IF . . . sometime soon after Zechariah’s return home from work Elizabeth conceives, that would place the 6th month of her pregnancy and the first month of Mary’s pregnancy ( Luke 1:23-41) somewhere around the month of August . . . IF . . . IF . . . that is the case according to the information available to us in the Word, that would place the birth of Messiah (Jesus – Yeshua in the Hebrew) somewhere around the end of May or first of June . . . about the time of Pentecost. . . in our humble opinion . . .
Also, the birth of Messiah at the Feast of Pentecost and the outpouring of the Spirit (the Ruach) at Pentecost ( Acts 2 ) would tie together the two halves, if you will, of the ‘promise of the Father’ . . . the birthing of the Seed of the Woman – The Messiah – ( Gen 3:15 ) and the ‘outpouring of the Holy Spirit in ‘the last days’. ( Joel 2:28-ff and Acts 2 )
We are commanded to remember His death until he comes again. Why ?
Many were born in the history of mankind.
Many were resurrected in the history of mankind.
Only ONE died and shed Blood with the Power to reconcile me to my Heavenly Father.
If I am reconciled by His death, how much more am I saved by His resurrected life. Romans 5:10

” I am crucified with Christ: nevertheless I live; yet not I, but Christ lives in me: and the life which I now live in the flesh I live by the faith of the Son of God, who loved me, and gave himself for me.” (Gal 2:20)

Hebrews 10:20 speaks of ‘a new and living way’. That is not exactly what the Greek says.
It speaks of a ‘new, fresh slaughter that lives . . .’
Only Yeshua could die and live again to reconcile me and save me.
Baruch Haba B’Shem Adonai – Blessed is He Who comes in the Name of YHVH.
Shalom  to you and yours –  B’Shem Yeshua HaMashiach – In the Name of Yeshua the Messiah