I do not know the exact source of that wiki article, and I'm not going to comb through Roddenberry biographies to find it. It is well established fact that Roddenberry was an atheist and he wanted his show to reflect his ideas.
Chaplains are part of an organised religion, even though some people might have vague 'oh, maybe there is something after death' type of ideas, you do not need chaplains for that. Certainly an open minded person consideres all possibilities, but that is not having a religion, nor does it take faith.
As for Katras and 'unexplained spiritual things' here exactly is the problem with religious/magical thinking and why it has not place in a scientific exploratory organisation. Such things just do not exist and cannot be assumed to exist. There are only things you understand or do not yet understand. People in Star Trek never resign in a face of an unusual phenomenon and say 'oh, maybe it is a miracle!' They encounter beings that are obviously gods, yet they do not treat them as having any greater spiritual significance. They are just powerful aliens. And they try to study them.
As for prayers, do you seriously think that they can alter the reality in real life? Because that obviously is not the case.