I think what is crucially left out here is culture, and in my opinion that culture is in fact not religion. Culture is how we "do" society, and not simply cultural touchstones such as high art, etc. It is the everyday, which I know from your writing you understand.

It is my position that culture and religion are not the same thing, though they are intertwined, quite inseparably. This is to say that culture is not a subset of religion, as you suggest.

It is in the aspect of belief in which this is most clear. We often find that the belief in the average person is not what is purported to be the religious belief of the established religious orthodoxy. Thus you have so-called non-practicing Jews or Catholics or whatever it is. One cannot escape culture though as it is the very stuff of society, so those non-practicing often have much in common with the practicing, and it is the culture which they share.