More specifically, a noncommercial freeform radio station.
Yes, for 35 years from 1982-2017, (with breaks for a couple of years in the mid-'90s and mid-'00s), I was a weekly host of the weekday 1960's/'70s show "Lost and Found" on all-volunteer non-commercial non-profit freeform WMBR at MIT. The show plays '60s/'70s music mostly not heard on commercial Classic or Oldies stations. I chose to model my edition from my memories of listening to WBCN's early freeform rock "American Revolution" format years 1968-1974.
This was my last Christmas/Winter Solstice show on 12/20/2016. This may have flown on WBCN over half a century ago, but not on any ratings-conscious commercial or public radio station since then. Some of these are not explicitly Christmas songs, but could be associated by a stretch of the imagination:
The Electric Prunes - "Agnus Dei (Lamb of God, Grant Us Peace)"
Fifty Foot Hose - "God Bless the Child" (cover of the Billie Holiday classic by obscure San Francisco '60s psychedelic band, their one album was produced by one of the Grateful Dead's longtime sound engineers)
The Bee Gees - "Every Christian Lion Hearted Man Will Show You" (from their first USA album)
Ultimate Spinach - "Fifth Horseman Of The Apocalypse" (local Boston area band)
Winter solstice, longest night of the year set:
The Youngbloods - "Darkness, Darkness"
War - "Slipping Into Darkness"
The Beach Boys - "Fall Breaks And Back To Winter"
Family - "Winter" - (by request) - (British progressive rock band)
Neil Young - "Winterlong"
Simon & Garfunkel - "Hazy Shade Of Winter" - (by request)
The Doors - "Wintertime Love" - (by request)
The Blues Project - "Steve's Song" (lyrics: "call me from a snow white, starlit stairway...", "the sounds of our winters love at nighttime...")
Blood, Sweat & Tears - "Sometimes In Winter" - (by request)
It's A Beautiful Day - "White Bird" - (by request) - (lyric: - "In a golden cage, on a winter's day...)
Pink Floyd - "Merry Xmas Song" - (by request) - (unreleased from 1969 BBC performance, available online)
Jethro Tull - "Christmas Song" - (by request)
Jimi Hendrix - "Little Drummer Boy / Silent Night / Auld Lang Syne" medley (by request)
Arlo Guthrie - "The Pause Of Mr. Claus" - (by request)
Martin Mull - "Santa Doesn't Cop Out On Dope"
Procol Harum - "A Christmas Camel"
Rotary Connection - "Silent Night Chant" - (by request)
Leon Russell - "Slipping Into Christmas"
Joni Mitchell - "River" - (by request) - (lyric: "It's coming on Christmas...")
The Grateful Dead - "Run, Rudolph, Run" - (cover of the Chuck Berry Xmas hit, only performed by the Dead live several times in Dec. '71, sung by their late original blues singer Ron "Pigpen" McKernan. He was too ill to perform by Dec. '72 due to a shot liver from heavy drinking, and passed in March '73. The Dead never performed the song again without him).
The Band - "Christmas Must Be Tonight" - (by request)
Greg Lake - "I Believe In Father Christmas" - (by request)
Some others I played in previous years Christmas/Winter shows (not in order of airplay):
Tom Rush - "Urge For Going" (lyric - "Winter's closing in")
Tom Lehrer - "A Christmas Carol"
Simon & Garfunkel - "7 O'Clock News/Silent Night"
Laura Nyro - "Christmas In My Soul"
Neil Young - "The Star of Bethlehem"
The Who - "Christmas" (from "Tommy)
Cheech & Chong - "Santa Claus & His Old Lady" - (by request)
Johnny Winter - "Please Come Home For Christmas" - (by request)
Mott The Hoople - "Death May Be Your Santa Claus" - (by request)
Captain Beefheart - "There Ain't No Santa Claus on the Evenin' Stage" (by request)
The Sonics - "Santa Claus"
Canned Heat - "Christmas Boogie"
The Christmas Jug Band (w/Dan Hicks) - "X-Mas Shopping Blues"
Commander Cody - "Daddy's Drinking Up Our Christmas" - (by request)
The Strawbs - "The Winter Long"
Bob Dylan - "Winterlude" - (by request)
Gil Scott-Heron - "Winter in America" - (by request)
Lindisfarne - "Winter Song" - (by request)
Jesse Winchester - "Snow"
Jethro Tull - "Ring Out Solstice Bells" - (by request)
John Mayall - "Chrismas '71" (sic)
Brian Auger & The Trinity - "We Three Kings"
The Kinks - "Father Christmas"
The Beatles - "Happy Michealmas" (etc...) - (by request)
Genesis - "A Winter's Tale" - (by request)
Gordon Lightfoot - "Song For A Winter's Night" - (by request)
The Rolling Stones - "Winter" (by request)
The Choir - "It's Cold Outside" - (by request) - (1966 hit by this Ohio garage rock band, some members became the Raspberries in the '70s).
Pentangle - "Child Of The Winter" - (by request)
Frank Zappa - "Don't Eat The Yellow Snow"
Captain Beefheart & His Magic Band - "Steal Softly Through Snow"
Steve Miller Band - "The Beauty Of Time Is That It's Snowing"
The Smothers Brothers - "Santa Claus"