To some extent probably, but I wonder if that's as true as you assume.
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.
Perhaps! I've encountered lots of fellow Christians who construct arguments by taking individual verses from everywhere in the scriptures and arranging them into an argument assuming that all the terms are equivalent and that the context doesn't matter. It particularly perplexes me when someone will cite, say verse x and then immediately after it cite verse x-2, reordering them in a way that destroys the sense of the passage, but supports some argument.
Perhaps there's an uncanny valley between people who are familiar with the broad strokes and people who argue contextually.
This practice is usually called "proof-texting" [1] and perhaps surprisingly it's not confined to one particular kind of Christian. For example, the very conservative evangelicals I grew up with were extremely anti proof-texting and decontextualized reading, to the point that they rarely read from the Bible (in church or otherwise) unless they're reading a whole chapter. On the other hand, the conservative evangelical Dwight Moody is often accused of promoting exactly this hermeneutical approach. In my experience, a certain class of the "reformed" are often guilty of it.
To be clear, I certainly think the people I grew up with were guilty of their own particular hermeneutical weaknesses, this is just about proof-texting.
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.