My point is that there is no superposition of dead and alive. The cat will cause the probability function to collapse long before we open the box. What am I missing here? Isn't the cat capable of collapsing the probability function? If not, why?
A theorem of von Neumann says that it doesn't make a bit of difference whether you model the cat (or anything else along the causal chain between closing the box and opening it to observe the cat) as capable of collapsing the wave function. You'll make exactly the same testable predictions no matter where along the way you place the collapse.
(The argument occupies the final chapter of this book, but recreating it is not a terribly difficult exercise.)
So feel free to posit that the cat collapses the wave function. Or to posit that only a human has that power. And if you prefer one story while your neighbor prefers another, let a thousand flowers bloom.
According to the Many Worlds view, no cat, nor even a human scientist inside the box, will "collapse the wavefunction". The scientist, the cat, and the radioactive particle are all components of the universal wavefunction, which simply branches when the radioactive particle both decays and does not decay. To an observer outside who can't see inside the box or interact in any way with what is in the box, the scientist both dies with the cat and survives with the cat, until the observer peeks inside the box. At that point, the outside observer's wavefunction branches because it has become correlated with the wavefunction of all that's inside the box. The idea that "consciousness" causes wavefunction collapse has no meaning. The scientist who subjects himself to the cat's fate is conscious (aware) in one branch that he survived; in the other branch he might briefly be conscious/aware that he is dying.
This is why I generally prefer a "mostly subjective" viewpoint of quantum mechanics as it is really, despite looking at all the alternatives, the only one that fits the closest to the mathematics of the theory as given with no other adulterations (other ideas like MWI, Bohm, etc. really are "different theories" in that they play with the maths and seem to have a fixation on eliminating and explaining away the collapse concept, and they thus really are "empirically equivalent theories" [except when they're not!]. My thinking has been that we need to take it blunt, at face value, and see where that leads.).
On a subjective account, the wave function belongs to you, the one outside of the box. It models, your information or knowledge about the state of affairs in the box. The transition from "live cat" to "live or dead cat" to "dead cat" starting from the initial state is just showing how your best knowledge, without looking into the box, that you can predict from that initial state, changes. All we can say about the "superposition" at the in-between point is that it means the predicted information about the answer to the question "is the cat alive?" is less than one bit.
That said, the theory does force us to admit that there really is something "odd" going on "in reality", otherwise it would just be doable with classical mechanics. But that "oddity" is more that the Universe seems to have an information limit that prevents the answers to all questions about a system from existing with perfect information at all times. That is, in some real cases, there really must be less than one bit of information, say, to some yes/no questions.
Problems identified with the subjectivity of the wave function in literature seem to be hung up on the idea that if you take it as subjective, you are taking it as subjective with some further assumptions on what the "real" reality should look like that often amount to sneaking classical mechanics in the back door, instead of letting the maths guide you as to what you can/can't say thereabout - which is that, if we make no such further assumptions, except perhaps relativistic causality, then you have to say that however it exists, physical parameters have "limited resolution" - limited, even fractions of a bit of, information.
In particular, the "real" information at the point in between queries ("measurements") must lie somewhere between the absolute maximum, as established by the Heisenberg uncertainty principle, and the level given by the Schroedinger equation, which may be somewhat less - the latter being, in effect, what @WillO's answer is one way of saying. There are endless scenarios we can imagine for the in-between that we would not be able to falsify.
As a subject, the cat may be assigned a wave function talking about the information it has regarding the contraption that is going to kill it. Of course, soon after that one "collapses" then there won't be any more wave function any more because this subject, the information-bearer, has been terminated.
Hence, from that point of view, it makes no sense to ask this question as it is posed because the wave function in question models your knowledge, not the cat's. The cat can't do anything to that. Well, maybe it can - it lets out one final scream as it dies, you hear that through the box, and update your knowledge accordingly :)
In my opinion such an experiment cannot actually be performed because the assumption that a box exists so that the macroscopic states of the objects inside the box is hidden from the exterior is false.
The state of the cat (dead or alive) has observable consequences outside the box, no matter what the material of the box is. For example, the gravitational field of a moving, breathing cat is different from the gravitational field of a dead cat. This will influence the way the objects outside the box move. If an external observer assumes that the cat is alive or in a superposition dead/alive but the cat inside is dead he will make wrong predictions about the motions of other objects he can observe.
So, in my opinion, the box is an impossible construct that has no consequences on what QM predicts. If the cat dies, it is dead for all observers, everywhere.