Earlier quoted context omitted.
Good observation, but I think it's overstated. I don't know many people who read the bible as a series of disconnected logical propositions, they mainly read it in much longer sections covering some topic. Chapter and verse come in when you want to quote something, or shorthand something, or refer someone to a specific and narrow section. In this usage, it's very similar to the way we cite Shakespeare, Homer, Chaucer…
I'm thinking, for example, of the RCL, which often breaks passages mid chapter and occasionally omits individual verses. What sort of Christian community do you mostly know? If you said "The St. Crispin's day speech" I'd immediately know what you mean. I'm definitely a Shakespeare fan, but would have no idea which act, scene, &c. that speech occupies. I'd wager that's much less true of the scriptures in many circles.
I'd guess that many people in vaguely Christianized communities would be familiar, to some degree at least, with passages like "Daniel in the Lion's Den", "The Sermon on the Mount", "The Plagues of Egypt", "The Good Samaritan", and so on, without being able to tell you the chapter and verse (or even book) in which they're found.