^^^ It would make sense that a ship's counselor could serve in this particular role, as it would probably be expected of them to acquire a proficiency in comparative religious or spiritual studies of various cultures (including human) as a part of their curriculum to be certified as such. And the 1701 DID have a chapel (Balance of Terror) - the more spiritually-inclined crew members would be free to pray to whomever they liked in that facility. If they needed guidance or interpretation in their respective scriptures and/or doctrines, the counselor could theoretically be trained to assist in that process.
The resources are all there; they're probably just not as recognizable by our standards. Meta-speaking, we've never really seen it, as the stories have never really called for it, but in various episodes, Troi has always come up with some nugget of wisdom about some particular culture-of-the-week that assisted in solving a particular problem. Why couldn't xeno-spiritualism also be in her bag of tricks?