I simply don't understand your comment that 'as you yourself point out, Christmas related themes don't really go along with the leaving of Rivendell'.
You yourself pointed out that the text doesn't really make this connection when you wrote, "However, none of these themes are explicit as the Fellowship leaves Rivendell quietly on the evening of what would later become Christmas Day and, '
with one glance at the Last Homely House twinkling below them they strode away far into the night'. "
The themes I mentioned were: Hope is born. ... Hope is born, as the counsels taken will work.
The impression I get from the departure is not so much hope as it is desperation and determination. Even the line as they leave of, "There was no laughter, and no song or music. At last they turned away and faded silently into the dusk," seems pretty far removed from the Christmas account.
A King is born. ...A King is born, as Aragorn will reclaim his throne.
This is probably the strongest connection, but it still seems rather tenuous to me, as very little attention is given at this point to the King heading out to claim his kingdom. In fact, I don't think I find any reference to Aragorn's kingship since Boromir's reference at the Council (to the sinews of the Kings of Men) until he obtains the Elfstone and the others mark how tall and kingly he stands.
A Star will guide. ...A star will guide, as Earendil's star guides Sam in Mordor.
What does this have to do with setting out from Rivendell?
The Wise will give gifts. ...The Wise will give gifts, as Galadriel gives the phial with the light of the Silmaril to Frodo.
Again, what does this have to do with setting out from Rivendell?
Angels will be present. ... Angels will be present, and there is Gandalf.
Angels are present throughout the Scriptures, and it's more their actions that are associated specifically with Christmas; not their mere presence. We don't see Gandalf doing anything particularly Christmas Angel-y, so this connection too seems rather weak.
Actually, since Gandalf has a sword, and Bill the pony can "nearly talk" as Sam puts it, we may as well say that Tolkien was meaning to refer to Balak's summoning of Balaam in Numbers 22 rather than to Christmas.
Shepherds (if shepherds there be) will rejoice and wonder.... Shepherds will rejoice and wonder, and the Elves (if Shepherds they be, contrary to the thoughts of Lindir) do rejoice and wonder.
Do they? The lack of laughter, song, and music during the departure doesn't sound very much like Elvish rejoicing and wondering to me.
Estel, estel, estel! The whole LOTR from this point on is replete with Christmas themes.
If by "Christmas themes" we mean, "very common themes that could vaguely refer to Christmas, or to many other stories," then sure.
As far as what chucking the Ring in the Fire has to do with the Annunciation? Everything! When the Ring goes into the Fire, Sauron is destroyed. When God incarnates himself as Christ in the womb of Mary, Satan is destroyed (or at least weakened - Men are redeemed). Sure, to JRRT, the Ring into the Fire is a lesser and just a foreshadowing event to the Annunciation, but it is similar!
Not really. If Tolkien were meaning to parallel the destruction of Sauron with the defeat of Satan or the redemption of Men, surely he would have chosen Good Friday or Easter rather than the Annunciation. Yes, the Incarnation of Christ was an important and awesome event, but it was the Resurrection that was the culmination of the redemption plan.
Furthermore, these particular dates would be an odd pairing, as the Annunciation occurred
before the birth of Christ. Having a journey leading from Christmas to the Annunciation makes no sense as a Christian message.
I find it frankly impossible to believe that bookending the quest with Christmas and the Annunciation is a coincidence!
And without evidence suggesting such, I find it impossible to believe that the dates are anything but coincidence.