I'm starting to like the idea of a gay alien atheist chaplain.
As to the question about Federation members refusing to serve on a ship without one? So what? My understanding is that Federation membership doesn't compel a planet to contribute to Starfleet's ranks... in fact, all indications are that Starfleet generally wants willing and highly competent individuals. AFAIK even during the Dominion War they never instituted a draft.
If the chaplain -did- have a form of spirituality that they ascribed to I can imagine crewmembers being quite put-off by the notion that that individual had the captain's ear in any capacity in which someone of their own spirituality did not.
I've gone to allegedly secular sleepaway camps that nevertheless had "optional" Chapel services and "non-denominational" Grace before meals and such...and I have to say I found it pretty uncomfortable. Even if you accept that they're not attempting to impose their beliefs on anyone, one has to wonder how they would have reacted if asked to make a more demonstrative effort to accommodate other belief systems.