Why has Schrödinger’s Cat, the experiment, not actually been performed?
Basically because we have no way of putting a system as complex as a cat into a pure quantum state.
Schrödinger’s cat thought experiment was intended to challenge the idea that a system can have undetermined values of some observables in a way that does not just correspond to a lack of knowledge on our part. If the killing of the cat were the result of a purely classical source of uncertainty such as a coin toss we could just say that the cat is either truly alive or truly dead and we just don’t know which until we look. But quantum mechanics includes uncertainties of a kind that cannot be interpreted as just the result of incomplete information.
An example on a scale much smaller than that of a cat is if a single electron has a known value of its vertical spin component (up or down), and if we subsequently divert it by a device that sends it in different directions depending on its horizontal spin component, then there is a 50% chance of seeing it go in either of the two directions and it appears (from an analysis of the observed probabilities in various other directions) that there was no way of predicting the horizontal component (say from some other initial observations) before we actually measured it.
Schrödinger’s point was that there’s something weird about this business of not having a property until we measure it, and he used the cat as an extreme example. But in practice, if we are good enough detectives, there will always be evidence in the box telling us exactly when the cat died; and so, although there may have been some time before we knew the outcome, we never see anything that looks significantly different from what might have happened if the killing of the cat had been triggered by some classical probabilistic event (such as say the first double six in a series of dice throws).
In order to experience the weirdness of having a cat that is neither alive nor dead (sometimes referred to as being both at once) we would need to keep the system of cat and triggering nucleus (and everything else that they can interact with) in what is called a ‘pure’ quantum state, which requires having complete information about the quantum states of all its constituent elementary particles.
This is obviously impossible for a cat, but experiments have been done in which larger systems such as complex molecules are put into superposition states where something like the shape of the molecule (which perhaps seems more substantial to us than the spin of a single electron) does not have a value until we observe it. These will undoubtedly get larger and more impressively weird seeming as technology improves, but I am pretty sure that they will never reach the scale of an actual cat.