The more fundamental problem is that quite a few of the cultural and intellectual assumptions Lewis made are not longer shared in our much more nihilist culture. He tried to start at the lowest level, but the "lowest level" in our culture is a lot lower now, and you cannot assume that people will recognize any connection between assent to some system of morality and any implications for personal behavior, for example. "If you say a modern celebrity is an adulterer, a pervert and a drug addict, all it means is that you've read his autobiography." P.J. O'Rourke
I remember trying to work out an explanation for sin that would be comprehensible to a modern student. The best I could do was "Have you ever done something that if a friend had done it, would hurt your friendship?" That's a bit clunky, and the hearer isn't likely to answer honestly anyhow.
I recommend C.S. Lewis to people who are already Christian but kind of shaky on the whys and wherefores. I tend to agree with Spufford that most non-Christians I know won't benefit much from his approach. I think that's unfortunate, but that's the world I live with--and even the one Lewis lived in too, if I interpret the passage from Screwtape correctly:
Are you not being a trifle naive? It sounds as if you suppose that argument was the way to keep him out of the enemies clutches. That might have been so if he had lived a few centuries earlier. At that time humans pretty well knew when a thing was proved and when it was not. And if it was proved they really believed it. They still connected thinking with doing and were prepared to alter their way of life as a result of the chain of reasoning. But what with the weekly press and other such weapons we've largely altered that. Your man has been accustomed, ever since he was a boy, to have a dozen incompatible philosophies dancing about together inside his head. He doesn't think of doctrines as primarily true or false but as "academic" or "practical" "outworn" or "contemporary," "conventional" or "ruthless," jargon, not argument is your best allie in keeping him away from the church...The trouble about argument is that it moves the whole struggle on to the enemies own ground...By the very act of arguing you awake the patient's reason, and once it is awakened who can foresee the result. Even if a particular train of thought can be twisted so as to end in our favor, you will find that you have been strengthening in your patient the fateal habit of attending to universal things and withdrawing his attention from the stream of immediate sense experiences. Your business is to fix his attention on the stream. Teach him to call it "real life" and don't let him ask what he means by "real."
I gather he wrote for the few who would listen, and did it well. Which is about the best we can do. The new generation needs some new introductions, from new writers faithful in their generation.