Look at your Constitution one more time.  Chaplains have been around from the United States' beginning, and on salary.

To your point, these are supported by private orgs, which are supported by donations.  They don't proselytize, nor do they get in the way.  The only thing you're paying for is transportation for them to get to a suicide/murder/rape scene at 2 a.m. to comfort those who want it.  

Unlike you, most do appreciate the comfort.  Considering all you're paying for is gas (believe me, that car will still be in inventory regardless of chaplains being there), that's a bargain.