I have not read all 16 pages of replies... so please forgive me if my following thoughts are missing the ball game...
I am an ordained presbyter, and my full time work is as a chaplain in a secular hospital. (I manage the department, actually.) My hospital has 325 beds for patients, plus ER beds, and a staff of approximately 2200 located on site (an additional 2000 are at off-site locations). In our instance, we have 4 full time chaplains, and 5 part time chaplains (myself included). If I pull out my religion census today... I have members of 38 different distinct religious groups (be it denominations within a larger tradition, or separate traditions) in our hospital, and, based on my experience, at least 80-90 different religious faiths represented among our staff.
In the twelve and a half years I have worked in this environment, I have had precisely five complaints about Chaplaincy existing. Three were complaints that their particular faith didn't have an employed chaplain on staff. Two were non-believers complaining that we existed at all.
My work includes several elements: first and foremost, I provide for spiritual needs of any faith tradition when requested. Now, that does not mean that I become a Roman Catholic if an RC needs a priest, and then a Muslim if the patient needs an imam. It means I am responsible for sitting with them with no agenda, and trying to help them figure out what can assist them in their time of need. When someone needs something Sacramental or faith-specific that I can provide, I provide it when invited to do so, and only then.
In addition to my duties as a Chaplain, I also run our morgue, handle advance directives, make death notifications, assist physicians and allied health professionals in devising morally and ethically appropriate treatments for patients with particular religious concerns (i.e., Jehovah's Witnesses refusing blood, etc.), handle bereavement duties, plan morale-boosting programs (like our quarterly sing-alongs), and other duties as assigned.
About 25% of my time is spent on arguably religious duties. Of course, I am a manager, so in my staff, it runs higher, over 60% in some instances. But our duties are vast.
Now, if you have a jack of all trades Chaplain like we deploy here in the hospital, that's all good and well, and it would be an option. The second option would be what has been previously suggested (and which I believe is the case in Starship Troopers) where the Chaplain is a person who is appointed for that work but holds a 'day job'.
We know that starships have chapels ("Balance of Terror" and "The Tholian Web"), so having a Chaplain to staff them isn't too much of a stretch.
The other option I can think of would be: Starfleet evaluates the religious makeup of a crew based on their documented preferences, and assigns certified Chaplains (either with another job or not) of the principal faiths represented to serve aboard the ship. If nine people can serve a hospital system my size, I would guess one or two could serve a Constitution class cruiser, and three or four a Galaxy class explorer.
Anyway, that's my perspective as a real-life Chaplain working in a secular environment. And yes, I work with pagans, agnostics, gays, bisexuals, athiests, and humanists on a regular basis with no real problems - mainly because Chaplains are trained not to push their religion, only to assist and enter deeper religious conversations when the door is open.