And why is their fall mentioned in the chronological context of Christ's (presumably) birth?
To me, it's not and takes place nowhere near the events of Christ's birth.
Long before that, Satan was in the Garden, not in Heaven...
So he had already fallen from his position as an archangel of God.
But he often visits the Lord ( Job 1:6-12, Job 2:1-6 ) to accuse His saints before the throne.
Why do I see it this way?
I believe that he was in the body of the serpent when the Lord had this to say of Satan:
"
and I will put enmity between thee and the woman, and between thy seed and her seed; it shall bruise thy head, and thou shalt bruise his heel." ( Genesis 3:15 ).
I also see that Jesus Christ has stamped ( and will stamp ) on Satan's head in a variety of ways...
First after His crucifixion when He rose from the dead,
then when He comes again to take the throne of His father David at Jerusalem for 1,000 years;
Finally when Satan is set loose for a short season at the end of the 1,000 years, deceives the nations into fighting one last battle, and then being stamped on for a final time when he is cast into the Lake of Fire.
But before that,
Jesus Christ shall be the "stone" that strikes the image and grows large, like a mountain, overtaking and replacing all the nations and empires of the earth ( Daniel 2:31-35, Daniel 2:43-45 ).
His millennial reign from Jerusalem.
That said, I take my leave of this thread and wish you well, sir.
My apologies for the long replies and for seeming to take over the thread,
as it was not my express intention.
May God bless you with much knowledge and wisdom in the days and years to come, George.