So you're saying chaplains can only "attend to" religious needs of soldiers if they share the same belief system? Then the chaplaincy program already has a really serious problem. There are any number of serious theological dogmatic divides among theists. Heck, even in the limited range of Christianity, there are serious divides over transubstantiation v. consubstantiation.
First, you're equating "atheist" and "secular", though the terms aren't synonymous. You're assuming all atheist needs are secular.
And, as I mentioned, you're ignoring the glaring differences among beliefs of theists when you make these pronouncements as to what an atheist cleric can or cannot do. In other words, even if we limit what a chaplain does to theistic faith needs, there's no requirement that the chaplain share any of the same god beliefs as the soldiers, so why require the chaplain to believe in any god or gods?
The only defense I've heard that makes any sense is that without a chaplain, soldiers who are stationed overseas or in combat would effectively be denied their First Amendment right to free practice of religion.
Of course that argument misses the fact that it would only justify chaplains in very limited circumstances (certainly not on U.S. soil, where access to churches and temples is easy and abundant). Also, if you believe that a chaplain of any faith can somehow supply these needs to soldiers of any faith, then there's no reason why a chaplain of no faith can't supply these needs to soldiers of any faith. In other words, if a chaplain is only a person who provides church schedules, bibles, korans, and such, then that position could be filled by anyone--not only a person with certain types of beliefs, or with credentials from a mainstream religion.
As you say, if the job is counseling or psychology, it's better to fill those jobs with people holding those credentials. (Many religions believe in faith healing. We certainly shouldn't hire "healers" to take over the job of medical personnel.)