Conflating religion and spirituality is a part of the problem (and I acknowledge that you did separate those). Religion is about glomming on to someone else’s spiritual experiences, while spirituality is about having your own. They may overlap at times, but one isn’t needed to have the other. Spiritual experience could be anything — the way you felt when you first saw your child or the way you feel when there’s a beautiful sunset. It doesn’t have to be overtly about God(s) or what you imagine soul is about.
For me, I feel most spiritual/soulful when I am working with clients and really helping them. I feel like my best, most authentic self and therefore in complete flow with the Universe (which to me is indistinguishable from the Divine).
My aetheist cousins go to a UU church, and at first I wondered why aetheists would bother with any church, but they express their connection to something larger than themselves by being active in social initiatives in their community. They just don’t believe in an old man in the sky (and neither do I). Connecting up with other like-minded people to help others is how they experience spirituality, even if that’s not what they might call it. A world without spiritual experience would be a dull and lifeless one, to say nothing of how lonely it would be. I know that “God is Love” is a really trite and overused phrase, but when extracted from the churchy expression of it, this is entirely correct. Love of all kinds is the ultimate spiritual experience! Just don’t get too hung up on the traditional words for things and you might find that as Peter Mayers says in his sacred science song, “Everything is holy now.”