There was at least one critic I have read who had a look at Jane Eyre and the references to The Book of Common Prayer... Other than that I haven't seen it so far, but then again, I was obsessed with the former and I haven't read so much about one book in particular since.
As for 'secrets', you could say that of the Catholic Church, although their theology or interpretation is not at all secret, although it is considered as a science that is taught at uni. The faculty now mainly contains lay people instead of priests and nuns (professors are even laymen). That said, though, theology is so vast and intricate that it is almost impossible to learn it without a proper book on it (like it is difficult to get any philosopher without works on the particular text and the general philosophy of the philosopher and his use of wording). Do physicists have secrets? I do not see people claiming that. Yet, not everyone knows how electricity works. Not everyone knows how The Holy Trinity (to take a Catholic example) works, yet you can find a theological explanation anywhere on the net if you want, just like the explanation about electricity, in the short and mor complex version. You can find even whole books on both.
As for protestants: they make a point of studying the bible themselves, daily, or at least that was the set-up. I don't suppose they all do it now, although there are bound to be some diehards still doing it. Where the Catholic Church kept the thing in Latin (unreadable for most people), the protestants translated it into the people's language in order to offer them the chance of studying themselves and they dragged their children to Sunday school to make sure they got the message of the stories, as did the Catholics with the Cathecism.
Whether that is good or not is not the point we are discussing. Fact is that the bible in itself is very flowery and contains a lot of old symbolism, like Medieval literature (not the same kind, but it is comparable in its mainly symbolica manner of writing). It cannot be just read and understood like a normal book. I mean, there might be people who believe that Moses heard a voice in the bush on Sinai (?) and then that God came down in a dark cloud; or that when Jesus died on Golgotha, God made it dark (it was probably a sun-eclips that lasted for a few seconds and not hours and hours on end) but is that really true?
But fact is that most people who wrote classics either went to church every Sunday (or even every day) or at least had read the bible (certainly in protestant countries) or the Gospel. They either loathed it and crusaded against it, or they really believed and made a point of proving it. It all depends on what experiences one has had with it.
Now may I ask, what is wrong with reading one psalm at the time? I haven't done it, only for Victor Hugo's Les Misérables, but that wasn't that bad. A few pages are not 300 pages of the Book of Kings or Chronicles! Or those few paragraphs of Joshua... Really, if you read it in little bits as you go along, it really isn't that bad.
Though I grant you the fact that it is pretty tedious to read it all in one go.