No I think you are very right. It's just all too easy for us, who inhabit a mostly secular public sphere, to apply our own mental/linguistic concept of "religious || non-religious" when thinking about cultures that probably did not apply such a rigid sort of filter to the world around them.
I guess that is a challenge in any kind of historical context.
I find it tempting to try to come up with better labels for the concept, despite it being unlikely that the greater anthropology community will ever shift to using them.
Spiritual zeitgeist might fit, I think—it describes the motivation behind most art that’s not associated with a particular coherent artistic movement. It’s “what people were feeling—and wanting to commiserate about feeling—at the time.”
I guess that is a challenge in any kind of historical context.