This might be a "metaphysical" question with no actual physics content and if such is the case then I apologize. Actually, to figure out whether or not it is an actual physics question is one of the major reasons I am posting this question.
Extremal black holes are the black holes that have the lowest possible mass for a given value of charge. If the mass were lower than the extremality value then the singularity would have been a naked singularity.
Now, naively speaking, quantum mechanics provides us with an attractive possibility of a black hole evolving into a naked singularity by the means of losing mass (while preserving charge) via Hawking radiation. But, as we know, when calculated, Hawking radiation ceases to exist precisely when the black hole becomes extremal - stopping the black hole from losing any more mass right at the stage at which it could have evolved into a naked singularity if it were to radiate even a little bit. This all seems extremely beautiful but even more freaky.
What I want to ask is that is there any reasoning due to which a Theoretical Physicist could have argued before actually calculating the explicit expression for Hawking temperature that it must be zero for the extremal black holes? Or do we have to accept this situation as a happy coincidence?
I think it is plausible that some might think that the censorship conjecture might be considered as the underlying reason behind this apparent coincidence. Thus, I would like to clarify that I think the censorship conjecture cannot provide any such reasoning as it has no deeper theoretical grounds but is just a form of wishful thinking. In more concrete words, the censorship conjecture just lays out some expectations that the consequences of all the fundamental laws should fulfill. It doesn't provide us with any understanding of the underlying physics. At best, it could have made the Theoretical Physicist hope that the Hawking temperature better be zero for extremal black holes. But it couldn't have provided rigorous grounds for predicting that it must be zero in the absence of an explicit expression for the Hawking temperature.
I think the fact that extremal black holes do not radiate should be considered as strong circumstantial evidence to the censorship conjecture. But the converse way of thinking the censorship conjecture as the reasoning behind the zero temperature at extremality seems absurd.
Note that naked singularities are not a problem in themselves. Because, in reality, there exists a well-behaved quantum gravity theory. Thus, it wouldn't be an internal inconsistency of the theory if naked singularities existed. We expect the censorship because otherwise, our experimentally verified calculations about the world that we have done without taking into account the effects of some naked singularities lurking around would have no reason to be verified in the first place. As far as I know, this is all the logic that we have behind the censorship. And it is an anthropic kind of reasoning. There is not any perfectly rigorous way of justifying my use of the word "coincidence" for the described situation but the following two are the reasons behind my use of the word "coincidence":
Charged black holes don't really exist in the universe. And thus, our anthropic conjecture shouldn't really be stretched to assert anything about them. But it surprisingly works even if we do such a naive thing.
Naively speaking, if we expect the censorship to emerge then also, zero temperature could have occurred at many other places including the extremality. But it happens precisely at the extremality and nowhere else. As if it specifically cared about the censorship. What I mean is that we ought to think of some basic principle of physics that very badly requires the censorship to be followed.