Screwtape Letters is actually pretty compatable with Judaism, if you overlook the Devil-as-evil vs. Devil-as-temptor-doing-his-job conflict (which is a pretty fine theological line, really) and which doesn’t come to play in the story in any meaningful way.
It’s the tale of a man struggling with both his religion and the conflict between concentrating on Earthly vs. Divine goals.
Not preachy Christianity, at all.
Favorite seen: the guy is sitting in church (could be a Shul), looking around and thinking “what a bunch of F!@ng crooks and hypocrites”. . . the Temptor encouraging certain thoughts.
You are right that anyone can relate to such thoughts. What about the thought that comes to your mind when an insufferable bore is regaling you with tales of her Perfect Child and how he just got accepted to Yale, bla bla bla... That's why we have Yom Kippur, to ask forgiveness for the horrible second of hoping the kid drops out and goes to Jamaica to study ganja growing...