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.
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.