I take it that your interpretation of the song is that the child had never been exposed to Jesus anywhere at all before seeing a crucifix (or whatever) in the church when her adoptive parents took her and that the crux of the story is that she magically knew who Jesus was without any stimulus, therefore "proving" that God exists.

Which is not what I took from it. It's hard to go through life in the US without being inundated with Jesus. I suppose as a child it may be fairly easy to not know what he represents, but you're still going to be familiar with the imagery. After all, he's almost always portrayed as a white guy with flowing hair and a beard, usually brown, wearing robes and sandals. Not exactly a common image otherwise. So the fact that she would recognize the image doesn't really mean anything to me. And that makes the song terribly bland from the "aww, cute" point of view and leaves any emotional impact with the hateful representation of the bad parents. The new parents are fairly blandly represented as giving hugs and going to church, where the song expends a lot of words (relative to the length of the lyrics) detailing how the bad parents were bad.

Actually, rereading the lyrics to check my facts, I see that I somehow missed that last stanza. So, to some extent, at least, nevermind. Cute story, definitely.

Actually, it's interesting to note that the song never states that the new parents attend church, either. It implies it, since the girl goes, but it never says that the new parents went or that they specifically sent her. (Picking nits, I know.)
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Bitt Faulk