I think there is some validity to this idea of an "inevitable impasse" when critical examination is brought to bear upon ANY set of beliefs to which a person has a strong psychological or emotional attachment. A person can, of course, have a strong emotional attachment in any set of beliefs, including atheism or a scientific theory. *In general*, however, I think this likely is the case more often with religious people than non-religious people.
When I say "religious" people, I don't necessarily mean all theists, I mean people who pray, go to church (or temple or whatever), join a mission, etc. I think it is generally true that religious people have a great deal invested emotionally in their religious beliefs. That is because the the very things that make them "religious"--gathering together in a community, repeating chants and singing songs, performing rituals and taking part in celebrations--all serve to reinforce the beliefs. People are taught at a very young age that all things good come from their religious beliefs--happiness, forgiveness, salvation, community, family, righteousness, etc. It is reinforced on a regular basis that (say) the existence of God is NOT just a fact of life, like the existence of the Sun, or the sky; no, the existence of God is ESSENTIAL for a person (you included!) to be happy, or imporant, or good, and if you believe in God like we do, then you get to be part of our wonderful community, and sing with us and have celebrations, and so forth.
Contrast this with a non-religious belief, say, the belief that light travels through a medium called the "ether". Near the turn of the 20th century, many physicists believed in the ether, which made a lot of sense at the time, before it was discredited by careful experiments. When it finally was discredited, no one said that life had no meaning anymore, there were no inspiring songs or ceremonies or celebrations that had to be abandoned; no one said that you can't be good without the ether, or you can't be saved, or forgiven, or go to heaven, or be part of a caring community who look out for each other, or that you can't be considered part of the same group as many of your family and friends, if you don't believe in the ether. No one felt a loss of identity, no one lamented, "The ether was a myth! Now I don't know what to believe anymore!"
But that is often precisely what devoutly religious people do if they become convinced that their beliefs are myths. You can offer a critique of a person's beliefs all you like, but even if those critiques are valid, if that person has enough invested in their beliefs you will never convince them. They aren't dispassionate judges of the evidence, they have a lot to lose (in their minds) if their beliefs are incorrect.
What I'm saying is of course true of anyone who has a lot invested in their beliefs. However--and I say this sincerely with no intention of being disparaging--I think this is less likely to be true, statistically, with an atheist than a theist. I for one will not see my entire world come crashing down if Apollo exists, I won't think there's no point in living anymore if there turns out to be a heaven, and so forth.