Bohr's original idea was that until there's some interaction it's pointless to assign a specific state - you can only talk about certain probabilities.
Schrodinger's cat is not both alive and dead. We know nothing about its current state, only how probable certain outcomes are.
I was taught that Schrödinger's Cat was conceived of as intentionally absurd, but that physicists loved it because it pointed out how absurd physics was being. As the other comment said, its not just that a photon might have gone through either slit, it provably went through both.
That's the case with any cat we throw in a box and ask somebody to kill it or not based on a dice, and then we visit the place where the box is and open it...
In other words, the cat is in a certain state, but we merely don't know which.
But this doesn't capture what the Schrodinger experiment is supposed to show. Such an case wouldn't need the experimental apparatus Schrodinger describes either.
The core idea of the experiment is that quantum uncertainty is not just of the "we just don't know yet" kind, but of the fundamental "reality is inherently mixed-up and uncertain / many things actually hold at the same time until observed" nature.
The purpose of the Schrodinger's thought experiment then is to show how this fundamental quantum uncertainty (i.e. not mere not knowing: actual "fuzzy" reality) "pollutes" the macro world (in this case, a cat).
The quantum-based mechanism that releases the poison, is his trick to transfer the uncertainty of the quantum state into the macro world (the box with the cat). (Funnily, Schrodinger's purpose was to discredit this).
Those that believe it's clear that "the cat is in one state or another, we just don't know which", haven't understood the problem.
You're arguing against what one might call an objective Bayesianism stance. That is to say that our probabilities reflect our inadequate information concerning an objective (classical) state of reality for the cat.
However the phrase
> Schrodinger's cat is not both alive and dead. We know nothing about its current state, only how probable certain outcomes are
suggests a perspective in which there is not an objective (classical) state of reality for the cat. Rather, probabilities might be seen as subjective assertions about the outcome of interactions with the system. It is meaningless to talk about the cat being dead or alive before looking. This is closer to what one might call a subjective Bayesian stance.
In actual classical reality, it's not at all meaningless to ask whether the cat is "dead or alive before looking". It might be uncertain, but it's not meaningless.
One might say "I think it's alive" and another "dead" -- and they have a chance to be right even before opening the box -- we'll just need to open it and see which was right ALL ALONG.
In quantum physics, on the other hand, in most interpretations, none of them is right before opening the box. If one says "the cat must be dead", and we open the box and the cat is indeed found dead, that person, wasn't right all along. He only became right upon the opening.
That's the whole paradox that Schrodinger tried to point out as an absurd result of quantum mechanics -- but which nonetheless is accepted by physicists not as a counter-example against QM, but as something still paradoxical that actually illustrates a key point about QM.