I was slightly surprised that they referenced the Easter story, until I thought about it a bit more. To be fair, his death is mentioned in quite a few carols, so there is precedent...
We three kings is blatant:
Myrrh is mine, its bitter perfume, breathes a life of gathering gloom, sorrowing, sighing, bleeding, dying, sealed in a stone cold tomb...
Hark the herald angels is more subtle:
Mild He lays His Glory by
Born that man no more may die
Born to raise the sons of earth
Born to give them second birth
The first Noel is probably between the two:
Then let us all with one accord
Sing praises to our heavenly Lord
That hath made heaven and earth of nought,
And with his blood mankind hath bought
The Holly and the Ivy is a bit more subtle, but references blood, thorns, and bitterness.
So there is an argument that some reference to the purpose of the birth is a not untraditional part of the Christmas story, though one that I suspect has lapsed in most nativities, so seems unusual for a non church related establishment.