If what we practice and learn in the dojo can be applied to the world outside of the dojo, then, yes, aikido has spiritual/religious/psychological value. Parents encourage their kids to play a sport because "it's good for them". It fosters teamwork and determination. Not just for physical exercise. Why do Christians go to church on Sunday? Just for the sake of going there? Ideally, they go to learn how to be better people.
The atheist's comments caught my eye. I'm more or less an atheist myself, but find value in aikido's philosophy. Learning to respect our partners on the mat, deflect aggression, and such have definitely increased my awareness of other people and how my behavior affects them.
Come to think of it, if church had more over the shoulder throws on Sunday, I would still attend.