Evidently Tolkien talked in a looser style than he wrote in
[In reply to]
As geordie has provided the video, and silverlode has retrieved the similar statement from the Letters (which is what I remembered too), it seems clear that when you said "it doesn't 'sound' like something he would have said", you were both right and wrong.
Tolkien did feel that death was a preoccupying theme in human letters - which might be what's surprising you, but that's Tolkien in a thoughtful mood, all right -- but now we see that he didn't usually write in the way he might speak off the cuff. The TV quote is almost charming in is casualness, and somewhat refreshing compared to his carefully balanced and moderate written prose. Now I wonder more than ever how his lectures sounded in the Oxford classrooms.