I really hate how it sounds like it's a new thing, since we know that the network (and even some of the animators) were on Charles Schulz's neck for the "And that's what Christmas is all about, Charlie Brown" bit back in the 60's. Yet, they managed to do oddly biblical stories on Animaniacs. I don't recall any religiously themed Looney Tunes shorts. Weird.Actually when the show started it might have been more allowed than it is now. I've seen episodes of Howdy Doody where the host talked about the audience going to church or synagogue. It certainly became controversial to mention God on a kid's show, but I dont think it was outright not allowed. I Could be wrong; I'd have to do some research on that.
The weirdest manifestation of that is the NBC broadcast of Vegitales. The network wanted to downplay the Christian values segment, which for better or worse (I guess technically) would find common ground on the morality Aesop of the series... if only they didn't wind up airing so many episodes based on actual biblical stories. How...how does that work? If you're hiding the religious content, why would you allow religious stories?
Then again, I remember seeing unaltered episodes of their sister show about Penguins. That's just... confusing at best.