This is an interesting question; I’m tapping into my recollections from my youth and singing in boys choir.
I think you have to bear in mind that most seasonal or “Christmas” music that we hear on the radio is divided into two genres: sacred music and secular music; most of what’s heard on the popular music radio stations is secular music that mostly caters to non-religious (and more specifically, non-Christian) listeners, that focus less on celebrating Christ’s birth, and more on secular ideas (the season, the feeling, shopping for and receiving toys, and of course, Santa Clause), and were for the most part written and recorded in early-to-mid the 20th century.
And in America, you hear a lot more secular music due to the influence of popular American music and the recording industry, and as an effect of holiday shopping; as radio and television gained in popularity in the 40s and 50s as many of the crooners and jazz greats wrote and performed songs that have since become part of our poplular tradition of the Christmas season in our culture. And, as this season is the high point for shopping and retail, this music becomes an element that is ingrained in most people’s minds-going to the mall and hearing these tunes while shopping.
But sometimes, you’ll also hear recording artists throw in a church hymn or two into the mix, and in some cases, they have not been translated into English (such as the Bach/Gounod hymn “Ave Maria” which is also a popular song in Bocelli’s repertoire, and also for other greats like Sinatra), and have seeped into the popular consciousness because of their recordings.
Most of the Latin hymnals that are related to the season of Christmas were written prior to recorded music, and intended for performance in churches (mostly, these songs were composed in the 17th-19th centuries). And, there are some popular hymns that are in French, German, and Italian (such as the French Il Est Ne Le Divine Enfant, or “Born On Earth the Divine Christ Child” and Silent Night, originally a Christmas Carol in German, Stille Nacht, heilige Nacht).
There are some really beautiful Christmas songs in Latin that are not circulated on popular music radio stations, and almost all choral works sung in Latin are written in honor of Christ, God, Mary, etc…and from these there are dozens of works that are sung around the Christmas season, even though for Christians/Protestants etc., most of this music is intended for performance year-round as part of the church service.
As for the English-translated hymns heard on the radio this time of year (as in your example “O come all ye faithful”), these were translated to English for two reasons: in recording industry, for the “crossover” factor of bringing these songs to the non-religious (secular) populations and contexts, and also for English-speaking churches such as the American Roman Catholic Church, the Anglican Church, and many of the Protestant denominations (Lutheran, Presbyterian, etc.), even though the latter are less inclined to singing hymns about Christ’s birth and the Passion of Christ, and are more focused on glorifying God in general.