And so this is Christmas....again, again.....

Although Trump is pushing me to say "Happy Holidays," just out of spite.

No, Daily Beast, just:  no.

Christmas has almost nothing whatsoever to do with "pagan" celebrations.

First:  Constantine set the date for Christmas?  I don't think so:

The December 25 feast seems to have existed before 312—before Constantine and his conversion, at least. As we have seen, the Donatist Christians in North Africa seem to have known it from before that time. Furthermore, in the mid- to late fourth century, church leaders in the eastern Empire concerned themselves not with introducing a celebration of Jesus’ birthday, but with the addition of the December date to their traditional celebration on January 6.

Besides, Constantine was dead by 337 C.E., which is another problem for the theory.  The biggest problem with it is:  history doesn't become simpler and less complicated the further back in time you go, and it isn't "made" just by the people you've heard of:

In Cyprus, at the end of the fourth century, Epiphanius asserts against the Alogi ... that Christ was born on 6 January and baptized on 8 November. Ephraem Syrus ....proves that Mesopotamia still put the birth feast thirteen days after the winter solstice; i.e. 6 January; Armenia likewise ignored, and still ignores, the December festival.... In Cappadocia, Gregory of Nyssa's sermons on St. Basil (who died before 1 January, 379) and the two following, preached on St. Stephen's feast (P.G., XLVI, 788; cf, 701, 721), prove that in 380 the 25th December was already celebrated there....

Please pause to note 380 C.E. is 43 years after the death of Constantine.  And as we will see later, some authorities note the first mention of the date (25 December) in Rome was in 354, or 17 years after Constantine's demise.

Jerusalem

In 385, Silvia of Bordeaux (or Etheria, as it seems clear she should be called) was profoundly impressed by the splendid Childhood feasts at Jerusalem. They had a definitely "Nativity" colouring; the bishop proceeded nightly to Bethlehem, returning to Jerusalem for the day celebrations. The Presentation was celebrated forty days after. But this calculation starts from 6 January, and the feast lasted during the octave of that date.... Cyril declares that his clergy cannot, on the single feast of Birth and Baptism, make a double procession to Bethlehem and Jordan.... He asks Julius to assign the true date of the nativity "from census documents brought by Titus to Rome"; Julius assigns 25 December.... But Julius died in 352, and by 385 Cyril had made no change; indeed, Jerome, writing about 411 (in Ezech., P.L., XXV, 18), reproves Palestine for keeping Christ's birthday (when He hid Himself) on the Manifestation feast. 

Lots of controversy over dates, in other words, well into the late 4th century.  I thought Constantine settled his much earlier, no?

Antioch

In Antioch, on the feast of St. Philogonius, Chrysostom preached an important sermon. The year was almost certainly 386, though Clinton gives 387, and Usener, by a long rearrangement of the saint's sermons, 388 (Religionsgeschichtl. Untersuch., pp. 227-240). But between February, 386, when Flavian ordained Chrysostom priest, and December is ample time for the preaching of all the sermons under discussion. (See Kellner, Heortologie, Freiburg, 1906, p. 97, n. 3). In view of a reaction to certain Jewish rites and feasts, Chrysostom tries to unite Antioch in celebrating Christ's birth on 25 December, part of the community having already kept it on that day for at least ten years....

Finally, though never at Rome, on authority he knows that the census papers of the Holy Family are still there. [This appeal to Roman archives is as old as Justin Martyr (First Apology 34-35) and Tertullian (Adv. Marc., IV, 7, 19). Julius, in the Cyriline forgeries, is said to have calculated the date from Josephus, on the same unwarranted assumptions about Zachary as did Chrysostom.] Rome, therefore, has observed 25 December long enough to allow of Chrysostom speaking at least in 388 as above (P.G., XLVIII, 752, XLIX, 351).

Constantinople

In 379 or 380 Gregory Nazianzen made himself exarchos of the new feast, i.e. its initiator, in Constantinople, where, since the death of Valens, orthodoxy was reviving. His three Homilies (see Hom. xxxviii in P.G., XXXVI) were preached on successive days (Usener, op. cit., p. 253) in the private chapel called Anastasia. On his exile in 381, the feast disappeared.

What?  No Santa Claus in Constantinople? Surely Constantine was turning over in his grave!

Rome

[This is where it gets very complicated, so let's just cut to the chase:]

In the West the Council of Saragossa (380) still ignores 25 December .... Pope Siricius, writing in 385 to Himerius in Spain, distinguishes the feasts of the Nativity and Apparition; but whether he refers to Roman or to Spanish use is not clear.... By the time of Jerome and Augustine, the December feast is established, though the latter ... omits it from a list of first-class festivals. From the fourth century every Western calendar assigns it to 25 December. At Rome, then, the Nativity was celebrated on 25 December before 354; in the East, at Constantinople, not before 379, unless with Erbes, and against Gregory, we recognize it there in 330. Hence, almost universally has it been concluded that the new date reached the East from Rome by way of the Bosphorus during the great anti-Arian revival, and by means of the orthodox champions. 

But certainly not before the end of the 4th century, and not at all because of "Saturnalia."  First:  yes, Christians did start adopting pagan temples as places of worship, pagan heroes as saints (I can think of a couple of Irish saints off the top of my head), and pagan practices as Christian:

This would change only after Constantine converted to Christianity. From the mid-fourth century on, we do find Christians deliberately adapting and Christianizing pagan festivals. A famous proponent of this practice was Pope Gregory the Great, who, in a letter written in 601 C.E. to a Christian missionary in Britain, recommended that local pagan temples not be destroyed but be converted into churches, and that pagan festivals be celebrated as feasts of Christian martyrs. At this late point, Christmas may well have acquired some pagan trappings. But we don’t have evidence of Christians adopting pagan festivals in the third century, at which point dates for Christmas were established. Thus, it seems unlikely that the date was simply selected to correspond with pagan solar festivals.

They did all that in the 7th century; well, to be fair, probably starting in the late 6th century.  2 centuries after the date for Christmas was finally established, however.

Saturnalia?  No.  That was on December 17 to December 23, but by 354 C.E. it was only on December 17.  It was more like the Feast of Fools than anything we think of as a Christmas observance (and remember we're talking about a religious ceremony here, not a civic one.  Christmas as we know it is an amalgam of medieval practices (feasting, mostly, anymore) and Charles Dickens' childhood memories (Pickwick and Scrooge are the roots of our modern Christmas, as well as Clement Clarke Moore).  Saturnalia held on as a holiday on the Roman calendar until the mid-5th century:

December 17 was recognized as the date of the Saturnalia as late as AD 448, when it was notated in the ecclesiastical calendar or laterculus ("list") of Polemius Silvius. But now, deprived of its pagan significance, it is identified only as feriae servorum ("festival of the slaves").

By then Christmas had been observed on December 25 in Rome for at least 60 years, so I don't think they were confused about the dates.  So maybe it was "Natali invictii"?  Well, maybe, as the date is the same as Christmas.  However:

But even should a deliberate and legitimate "baptism" of a pagan feast be seen here no more than the transference of the date need be supposed. The "mountain-birth" of Mithra and Christ's in the "grotto" have nothing in common: Mithra's adoring shepherds....are rather borrowed from Christian sources than vice versa.

Whatever link there is doesn't really mean anything.  Oh, and about the yule log, I'm not going beyond this:

The yule log

The calend fires were a scandal even to Rome, and St. Boniface obtained from Pope Zachary their abolition. But probably the Yule-log in its many forms was originally lit only in view of the cold season. Only in 1577 did it become a public ceremony in England; its popularity, however, grew immense, especially in Provence; in Tuscany, Christmas is simply called ceppo (block, log...). Besides, it became connected with other usages; in England, a tenant had the right to feed at his lord's expense as long as a wheel, i.e. a round, of wood, given by him, would burn, the landlord gave to a tenant a load of wood on the birth of a child; Kindsfuss was a present given to children on the birth of a brother or sister, and even to the farm animals on that of Christ, the universal little brother....

Not too many pagans running around Europe by the 16th century, really.  And what's left?  The Christmas tree?  Oh, please; I'm not going to repeat myself on that.  The twelve days of Christmas, where did that come from?  (It is the span of time between Christmas Day and Epiphany, or January 6.)  Nobody knows, but I'm comfortable saying;  not from pagans.  Certainly not from a yule log that burned for 12 days (unless you were a lucky 16th century English tenant!).  No, that's too simple; so let's start here and work backwards a bit:

The reason for the fixing of this date it is impossible to discover. The only tolerable solution is that of Mgr. Duchesne ..., who explains simultaneously the celebration of 6 January and of 25 December by a backward reckoning from 6 April and 25 March respectively. The Pepyzitae, or Phrygian Montanists, says Sozomen..., kept Easter on 6 April; hence (reckoning an exact number of years to the Divine life) Christ's birthday would have fallen on 6 January. But, it may be urged, the first notice we have of the observance of this date, refers to Christ's Baptism. But this (if we may assume the Basilidians, too, to have argued from 6 April) will have fallen on the exact anniversary of the Birth. But why preeminently celebrate the Baptism? Can it be that the celebration started with those, of whatever sect, who held that at the Baptism the Godhead descended upon Christ? On this uncertain territory we had better risk no footstep till fresh evidence, if such there be, be furnished us.

Indeed.  But working backwards, as I said, the New Advent articles tells us that Epiphany as a celebration of the church began in the Eastern church, where the idea of what actually was the "epiphany" wasn't settled for some time.  The baptism, the miracle at Cana (the first miracle in John's gospel), the magi, were all considered events of the epiphany of the divinity of Christ, and each was celebrated.  The shorter version is that the Eastern church attached importance to the Epiphany and celebrated it on January 6, while the Roman church celebrated the nativity on December 25.  Both could technically be epiphanies (revelation of the godhood and incarnation).  And sooner or later, the two were adopted on either side:

It is simpler to say that, about the time of the diffusion of the December celebration in the East, the West took up the Oriental January feast, retaining all its chief characteristics, though attaching overwhelming importance, as time went on, to the apparition of the Magi. 

Epiphany and Christmas were being celebrated long before the church reached deep into the lands where people were burning logs to keep warm in winter or worshipping trees or counting sparks from fires (which is kind of a stupid assertion, anyway, made by someone who's spent little time by a wood fire).  I'm comfortable in saying the dates were set long before the "yule" was named a log.

The term Yule is of disputed origin. It is unconnected with any word meaning "wheel". The name in Anglo-Saxon was geol, feast: geola, the name of a month (cf. Icelandic iol a feast in December).

As so many things are.  And Anglo-Saxon was not a language of Christians until quite a bit later than the 4th century, when most of these dates were being set.