While many people wouldn't have a problem with it, if it were done carefully and not too heavy handed, others would cry foul about any portrayal of religion in Trek. Trek fans can't even come to a consensus about Bajor and their religion. Portraying human religion on a Federation Starship would likely cause a sun eclipsing outrage.
Bry Sinclair makes some good points as well. How does one person handle the religions of 150 member worlds? It's one thing where a world has just one religion, but what about planets like Earth with multiple religions. How many people do you know that are well versed in multiple religions in addition to the one faith they practice personally. Granted, it's different when it's your job, but it's still a monumental task to take on.
One possibility could be some sort of non-demoninational service that is more about a general belief in faith and spirituality, rather than the worship of any specific God. You take all the benefits of a counselor, but come at it from the point of faith in a higher being of some sort.
Here's a question though. Where does this chaplain fall in the chain of command? Could they serve as an adviser to the CO as Troi did to Picard or are they just another civilian along for the ride...or a third option, an actual billet to be filled by an Academy trained officer or enlisted crewmen.