The problem is that this is not what Schrodinger predicts, though. If |alive> and |dead> are valid solutions to the Schrodinger equation of some system, then so is any linear combination of them whose norm is one, such as 1/sqrt(2) |alive> + 1/sqrt(2) |dead>. So the math of QM predicts that there should be infinitely many cats that should feel partly alive and partly dead, many times more than the one live cat and the one dead cat. And yet you don't find this in any classical experiment.
1/sqrt(2) |alive> + 1/sqrt(2) |dead> is a valid solution to the Schrodinger equation. That cat is half alive and half dead.
That is different than |A cat that feels half alive and half dead>, which is its own completely different state. For a cat to feel like it is half alive and dead, or to notice that it is half alive and half dead, there are distinct things that happening in its brain that don't happen in either the |alive> or |dead> basis vectors, let alone any linear combination of the two.
We could even put all those states in a superposition!
1/sqrt(3) |alive> + 1/sqrt(3) |dead> + 1/sqrt(3) |half alive and half dead>.