We've seen Counsellors but so far no chaplains, as far as I know, have been depicted in any incarnation of Trek. Seems to me like too much of a departure.
Realistically, you have two options: depict an extant religion, or a fictional one. Neither prospect excites me. Depiction of an extant religion won't be able to get in-depth without courting more controversy than networks will sign off on; the blandness and superficiality that will (I think) inevitably result pushes it into "Why bother?" territory.
On the other hand, a fictional religion seems no more interesting. Granted, you can get more in-depth, in all likelihood taking elements of real religions and cloaking them in enough 'fictionality' to deal with them in depth without provoking letter-writing campaigns. But the idea just doesn't excite me. The few depictions of spirituality in Trek that do exist have always seemed bland and uninteresting to me. It's always some reported
personal difficulty; McAlien's duty dictates that he do X, but (gasp!) McAlien's religion forbids him to do X! And then we get some hellishly long shot of McAlien staring at a little statue or whatever with strings in the minor chords on the soundtrack and then... McAlien either does or doesn't do X. The controversy over doing which is entirely informed and never connects viscerally with the audience. Yawn
, as far as I'm concerned. Obviously just my personal preference, but I'd rather we didn't bother doing that. It's worked well enough so far without that.