I would love to see a Chaplain in Trek. It would bring a modern day issue to the franchise that has not been seen before. Imagine the stories of conflict it could provide. Especially religion over science to explain the unknown.
Star Trek has had shows where the issue has been approached. Usually the religion is seen as superstition and scientific reality is the truth. 'Who Watchers the Watchers' is a good example.
Bajoran religion is portrayed in more positive light. Their gods are real and Federation sees them as an advanced sort of an alien. Still, Bajorians revering them is not shown as purely a bad thing, even though more dogmatic side of their religion sometimes is. Still, I'm sure Picard wouldn't approve, afterall, it's same thing as Mintakans worshipping him or if Federation started to worship Q. If Federation people truly were religious, they would approach prophets and other god-aliens differently. They would consider the possibility that these obviouly divine beings would have important spiritual messages for them. They don't. Were Starfleet to meet Jahve they would classify him as an non-corporeal energy being and ask him to explain the holocaust at Sodom and Gomorrah.
Regarding the Katra side discussion. Perhaps it isn't a case of the Katra is a spirit or it is a scientific process the Vulcans understand. Perhaps it is something the Vulcans cannot explain. They are ware of it and how to collect and preserve the spirit but not actually what it is. There was no evidence to suggest that they had any doctors who knew what was happening. How could you scan the Katra with a tricorder? It probably doesn't even read as a second brain wave on tricorder scans.
That it exists - but there is no explanation makes it either a pre-paradigmatic psi-science or a spiritual oddity.
Why people assume Vulcans don't understand Katras? It is represented as mysterius to evoke suspense in the movie, but whole Vulcan telepathy thing is quite odd from modern perspective, but that's not magic either.
I found it troubling that people assume there could be some sort of special sort of mysterious truths that cannot be studied by normal means. That is just not rational. Either so called 'supernatural' things are real, and then can be studied as any other phenomenon, or they are not real and are just imagined. There is no other categories, either in reality or in Star Trek.