Silver Bells does mention Christmas. So do Grandma Got Run Over By A Reindeer, Here Comes Santa Claus, Santa Baby, Most Wonderful Time Of The Year, Rudolf The Red-Nosed Reindeer, Ding-Dong Merrily On High and some versions of The Holly And The Ivy.

So does The First Noel (except it uses the French word).

The Wassail Song does not mention Christmas, and I'm not sure it's really a Christmas song any more than Deck The Hall is (or was originally).

Songs that I'm not allowed to sing at one of the Jewish Retirement Residences where I perform, because some residents consider them "Christmas songs," but which actually have nothing to do with Christmas (either with Saint Nicholas or the birth of Jesus) are:

Frosty the Snowman
Jingle Bells
Let It Snow, Let It Snow, Let It Snow
Sleigh Ride
Winter Wonderland
Marshmallow World In the Winter
Jingle-Bell Rock
Baby, It's Cold Outside
I've Got My Love To Keep Me Warm
Susy Snowflake



BTW, I believe I've read that "Jingle Bells" was not associated with Christmas until around the 1930s or 1940s, although the original song was from around 1850.