There is a sense in which the whole problem of the existence of supernatural powers such as God is soluble. You try to come up with the best broad picture of the universe. You pay special attention to phenomena that have traditionally been explained as due to gods and demons, spirits and ghosts. If a naturalistic picture ends up doing the best job and holds the best promise for further progress, you say that the gods are most likely not real.
There is another sense in which the problem is insoluble. Everything in debates over the gods tends to be contested. Many believers think that the broadly scientific approach I just described is not appropriate for this sort of problem. It becomes very hard to find common ground while seeking a solution, since whatever common ground we think we have will crumble if religiously-significant ideas ever come under threat during the debate. The goalposts are always on the move. The debate is constructed to systematically undermine the possibility of agreement between the faithful and the skeptical.
So unsurprisingly, whether one believes or not depends closely on personality and temperament. This in itself is no big deal. For example, you need the right sort of brain to understand physics. It’s very arduous and difficult to train people to have the appropriate temperament and outlook for doing physics. But also, in doing physics, we can plausibly set aside personal and political matters to engage in debates over physics. In discussing religion, this is not the case. Very often the debate is precisely over a person’s deepest commitments. The threat is that without God, a whole moral perspective and way of life will unravel. And this threat hovers over the whole debate in a way that does not happen in even the most heated discussion of physics.