Actually, I think what you are saying is very valid, and in fact I've had similar thoughts. If the cat walks in as a complicated mixed state of slightly different alive cats (the differences being too small for us to detect), then it never enters a superposition of alive and dead, instead it enters an even more complicated state-- a mixture of superpositions that project onto the cat subspace as something alive or dead. It's a mixture of superpositions that project onto a mixture! So then the question is, does a mixture of superposition states have the essential character of a mixed state or a superposition state? That depends on which one has the greater impact on measurements that we might do on the new system. The cat-mixture has very small differences in all those alive cats, but a decayed nucleus is an even smaller difference still. The cat paradox doesn't really work unless something really amplifies that tiny difference between the decayed and undecayed nucleus to the point where it can kill a cat (like a gun), whereas there is nothing happening in the cat that would otherwise cause it to die. So the cat mixed state matters less than the amplified state of the nucleus, so it might seem like the result is more like the superposition state than like the mixed state.
But a question remains-- what happens in that amplification process? This I feel is a crucial issue, because if we put a gun in there that can shoot the cat, then it is really the gun that we must wonder about, not the cat-- if the gun is in a superposition state of shot and unshot, then the cat will be in a superposition state of dead and alive. But how did the gun get into that superposition state? A nucleus can't make a gun fire, so what amplified the outcome?
So it is the amplification issue that is at the heart of the cat paradox, and it is also at the heart of the collapse of a measurement when you read a pointer that says if the nucleus decayed or not. Schroedinger is just asking the question, why don't we ever perceive classical pointers as being in superposition states when they are coupled to quantum systems? The equations of quantum mechanics suggest that if we have a closed system that couples a macro system to a quantum superposition, and amplify the importance of the quantum superposition, then the macro system should take on the character of that superposition.
But for some reason, amplification just doesn't do that, or at least we don't perceive it to do that-- instead, something in the amplification process confers the properties of the macro mixed state onto the quantum system, not the other way around. In a sense, our effort to amplify the importance of that quantum superposition has been foiled by the mixed state we coupled it to, and the superposition was somehow "lost in amplification." That would be the Copenhagen perspective, anyway, but I think this line of thinking is fruitful because we might actually be able to actually track how the superposition was lost in amplification, which might adjudicate between the various interpretations in a way that is currently not a part of the theory of quantum mechanics.
