The answer to the linked question is misleading (and I've left a comment there). You don't get to choose between nonlocal and non-real. While experiments violating Bell's inequalities do imply that no local, hidden-variables (i.e. real) theory is consistent with QM, that theorem must be taken in context with an earlier argument it's supposed to be a reply to, namely, the famous paper by Einstein, Podolsky, and Rosen (EPR). That paper argues that local QM theory implies that there are hidden variables (not quite the same as saying that local QM is always realistic, but it is in the context of entangled states). Taken together, the logical conclusion is that no local theory consistent with QM works - you're stuck with nonlocality.
You still have a choice between realistic and nonrealistic theories (or perhaps theories that are not completely realistic but do contain some hidden variables), but to produce predictions consistent with Bell's inequalities they will have to be nonlocal. The classic examples are Bohmian "pilot wave theory", which is realistic but highly nonlocal, and traditional QM, which is nonrealistic, but with its nonlocality apparently (?) being limited to some esoteric situations like particle pairs being generated in entangled states. Note that both those theories give the same predictions, so in that sense they are indistinguishable. That means it's probably an empty question to ask whether "reality" is realistic or unrealistic, or at least you can't tell based on those theories, since both the realistic theory and the nonrealistic theory provide equally good representations of reality.
To answer your question about why scientists have settled on the unrealistic theory (i.e. QM), it's because, compared with the only viable realistic theory (so far), which is Bohmian mechanics, it was developed first, it's easier to use for computation, and it has proven highly useful and successful. Bohm never intended pilot wave theory for everyday use; he just wanted to demonstrate that a realistic theory could be made.
I am not sure to understand well the issues you raised. I am sure that some of the following comments are already included in your questions.
I stress that the KS theorem and the Bell one have a very different nature.
The KS theorem does assume part of the quantum phenomenology and theory of quantum observables.
Taking advantage of Gleason's theorem -- which assumes the orthomodular structure of the lattice of quantum observables -- it proves that, for a quantum system whose Hilbert space has finite dimension and larger than $2$, its phenomenology cannot be described in terms of a realistic non-contextual hidden variable theory. It happens provided the outcomes of compatible observables satisfy some natural functional relations.
The choice between realism or non-contextuality is made by the standard interpretation of quantum theory, if one assumes it in toto. Indeed, the outcomes of measurements of an observable in QT do not depend on the choice of any other compatible observable simultaneously measured: QT is non-contextual. On the other hand, realism is not valid in standard QT, since the values of observables are not predetermined.
In principle however there could exist a realistic contextual hidden variable theory compatible with the quantum phenomenology, different from the standard interpretation of QT. The Bohm interpretation is considered such from some perspective.
Differently from the KS theorem, Bell's theorem does not use any quantum description or assumption. It proves that the measurements of some properties of a system made of two causally-separated parts must satisfy a certain inequality. It happens provided some realism and locality assumptions are satisfied.
There is no possibility to decide which assumption does not hold in case these inequalities are violated in the framework of Bell's proof.
Quantum systems violate those inequalities and, as before, the most common interpretation of quantum theory is that realism is violated and locality is safe. However, in principle there could be another hidden variable theory compatible with the observed violation which is realistic an non local or even non realistic and non local.
Personally, like most physicists I guess, I lean towards the validity of locality in spite of realism, but I don't think the debate is really over.
Best Answer
Realism refers to a philosophical position that says that certain attributes of the world of experience are independent of our observations. Let's take a physics example. In classical physics we used to say that a particle has a definite position and a definite momentum at a certain instant of time. These are represented by real numbers and they have those definite numbers independent of any observation. This seemed to be the only sane position one can take about the objective world. However the uncertainty principle of quantum mechanics tells us that a particle can not have both a well defined value of position and well defined value of momentum at the same time along the same direction independent of measurement. The more accurately one tries to measure one the less accurately one can have the knowledge of the other. Philosophically it means that position and its conjugate momentum can not have simultaneous reality. This realization had led the founding fathers of quantum theory to reformulate mechanics into a new theory called quantum mechanics. In QM a system is represented by a state vector in an abstract space. The length (norm) of this vector remain unchanged but with time its direction changes (for simplicity I am discussing Schrödinger picture). The various components of this state vector along the axes are various eigenstates with definite value of certain observables. Obviously the state vector is the linear combination of these eigenstates. Whenever a measurement is performed the state vector collapses to one of the eigenstates with certain probability determined by the Schrödinger's equation.
The so-called realists claim that the system was already in a definite state characterized by some additional hidden parameters before the measurement and since we are not aware of those hidden parameters we have an incomplete knowledge of the system. The random outcome reflects our incomplete knowledge of the system. There are number of hidden variable theories developed which reproduced the results of ordinary quantum mechanics.
Then surprisingly Bell discovered the famous Bell's inequality and showed that not all results are identical for both qm and local hidden variable theories. Experiment carried out and the verdict was clear. QM won. Nature supported QM. Therefore local hidden variable theories were ruled out. However there are nonlocal hidden variable theories which still survived like Bohmian mechanics. (I would also like to emphasize that MWI is an interpretation which is to some extent realist in spirit and it is by no means ruled out)
But what is locality? Locality is the assumption that an object can be influenced only by its immediate surroundings by the events which took place in its immediate past. All classical and quantum field theories depends on this assumption in an essential way. Non locality implies that two events which are separated from each other by space-like separation can affect each other. Some people demand (imho) falsely that EPR type entanglement violates locality. In reality in never does. All one need to abandon is realism. Entanglement just shows that there exists quantum correlations between particles which were in past had some common origin. It also shows that if it were a classical world then the EPR entanglements effects were nonlocal. But we live in a quantum world and there is no non locality.
Therefore in a nutshell, locality is certainly not ruled out. Realism is ruled out to a large extent.