I think the concept is sound, but I don't think Trek, at least past Trek, could pull it off properly. TNG had a hard enough time trying to write well for a counselor, even with added perks like empathic powers and political royalty to the character. Even DS9 made Ezri more of a counselor-in-title only.
Part of the problem, I think, is that Trek has rarely been character-driven, so there'd be less reason for characters to seek personal advice. It became easier for DS9 and Voyager to have those types of stories, not just because of different writing mindset, but DS9 had a large cast and Voyager depended on the "family" trope (so in a sense, everybody filled the role of counselor/chaplain in certain capacities).
So it's not outside the possibility for future Trek to have a chaplain, but I'd rather that the writers don't have one JUST for the sake of having one. There should be a narrative purpose to it (could you imagine having one aboard during TFF?).
On an semi-unrelated note, after watching seaQuest and Last Resort, I wonder why there's no equivalent to Chiefs of the Boat. Sure, there's O'Brien, but his place outside the chain of command as an enlisted crewman was rarely explored.