Historically, Christmas time meant mischief and drinking.
But without the business of gift-giving that sprang up in the 19th
century, Christmas might still be what it once was for many people: a
riotous bacchanalia in which drunken gangs brawled in the streets and
bashed their way into houses demanding money and alcohol.
(From "The Capitalists Who Saved Christmas" by Jason Zweig, WSJ, December 26, 2020, page B4.)
In the 16th through the early 19th centuries, according to the Oxford
English Dictionary, “merry” was a synonym for “drunk.” That
connotation still lingered when, in Charles Dickens’s “A Christmas
Carol” (1843), two men “wished each other Merry Christmas in their can
of grog.”
(op. cit.)
So at least in the 16th c., people were getting merry (= drunk) at Christmas. As can be seen by the Dickens reference, this historical usage persists on the American side of the pond.
But you might object: Isn't there plenty of drinking on New Year's Eve? Again, history plays a role.
A prominent educator and patron of the arts, Henry Cole travelled in
the elite, social circles of early Victorian England, and had the
misfortune of having too many friends.
During the holiday season of 1843, those friends were causing Cole
much anxiety.
Cole hit on an ingenious idea. He approached an artist friend, J.C.
Horsley, and asked him to design an idea that Cole had sketched out in
his mind. Cole then took Horsley’s illustration—a triptych showing a
family at table celebrating the holiday flanked by images of people
helping the poor—and had a thousand copies made by a London printer.
The image was printed on a piece of stiff cardboard 5 1/8 x 3 1/4
inches in size. At the top of each was the salutation, “TO:_____”
allowing Cole to personalize his responses, which included the generic
greeting “A Merry Christmas and A Happy New Year To You.”
It was the first Christmas card.
(From "The History of the Christmas Card", by John Hanc, published at SmithsonianMag.com.)
TL;DR
"A Merry Christmas and A Happy New Year to you" were on the first Christmas Card and have stuck.