Indeed, Pullman's tale tells of John Calvin assuming the papacy and moving the headquarters to Geneva, thus combining the Catholic and Reformation traditions into one.  In the movie, the Magisterium appears to be located in London.  In any event, the point is not subtle.

The most direct attacks upon Christianity and God do not appear until the last book, The Amber Spyglass, in which Lyra and Will (a boy her age who first appears in the second book) eventually kill God, who turns out to be a decrepit and feeble old imposter who was hardly worth the killing.

Is Pullman's attack on Christianity exaggerated by his critics?

No -- his attack is neither hidden nor subtle.  The entire premise of the trilogy is that Lyra is the child foretold by prophecy who will reverse the curse of the Fall and free humanity from the lie of original sin.  Whereas in Christian theology it is Jesus Christ who reverses the curse through His work of atonement on the Cross, Pullman presents his own theology of sorts in which the Fall is reversed through the defiance of these children.  As Pullman insists, Eve and Adam were right to eat the forbidden fruit and God was a tyrant to forbid them the fruit of the Tree of the Knowledge of Good and Evil. 

The supernatural element of Pullman's story is "Dust," which is seen by the Magisterium as original sin but is presented by Pullman as the essence of life itself.  In The Golden Compass, Lyra is given an "alethiometer" or "golden compass" which is filled with Dust and tells the truth to one qualified to operate it.  Readers are told that a great battle is coming in which forces fighting for human freedom and happiness will confront (and destroy) the Magisterium and God.

In the last volume of the trilogy, a character known as Dr. Mary Malone explains her discovery to Lyra and Will:  "I used to be a nun, you see.  I thought physics could be done to the glory of God, till I saw there wasn't any God at all and that physics was more interesting anyway.  The Christian religion is a very powerful and convincing mistake, that's all."

Is there more to the larger story?

Yes, and it has to do with sex.  Surprisingly graphic and explicit sex.  Pullman believes that the Christian church is horribly repressive about sex and that this is rooted in the idea of the Fall.  As he told Hanna Rosin of the Atlantic Monthly, "Why the Christian Church has spent 2,000 years condemning this glorious moment, well, that's a mystery.  I want to confront that, I suppose, by telling a story that the so-called original sin is anything but.  It's the thing that makes us fully human."

Puberty is a big part of Pullman's concern.  Coming-of-age stories are one of the most common forms of fiction, but Pullman's packs a punch that readers cannot miss.  He wants to celebrate the adolescent's arrival at sexual awareness.  Remember that the child's daemon can change forms until puberty.  At that point it is fixed as a single creature that reflects the personality and character of the young adult.

Puberty means the coming of sexual feelings.  The Magisterium would prefer that children grow up without experiencing sexual temptation, so it is conducting an experiment in order to separate children from their daemons before puberty, when their daemon can no longer change.  This procedure, known as "intercision" makes the child a "severed child" who has no daemon -- and thus no soul.  The Magisterium has assigned Mrs. Coulter the job of abducting the children and taking them to the North for this experiment.

As Mrs. Coulter explains to Lyra (who is revealed to be her own daughter) in the first book:  "All that happens is a little cut, and then everything's peaceful.  Forever!  You see, your daemon's a wonderful friend and companion when you are young, but at the age we call puberty, the age you're coming to very soon, darling, daemons bring all sorts of troublesome thoughts and feelings, and that's what lets Dust in.  A quick little operation before that, and you're never troubled again."