Just based on what you've written here, it's a little difficult to get a complete impression of what your story's about. Even still, I think I see what you're getting at, so I'll take a stab.
If I understand your question correctly, you expect that readers will identify with the society surrounding your protagonist, and you're worried that if you present content that criticizes this society, readers will feel insulted, and put down the book. Although, I wonder if this is actually the reaction people will have to your story.
The protagonist walks through life desperately needing help, and getting none. I do this in a way that is quite passive aggressive, initially painting society itself as the bad guy, though there is a villain.
Elsewhere, you've written that your story is told in first-person by your protagonist. This being the case, it makes sense that society would initially be painted as "the bad guy," as your narrator probably believes society truly is the bad guy. Most readers are going to intuitively identify with the narrator, especially if they are presented as a victim of other characters. So, unless you give them a compelling reason to do otherwise, readers will likely adopt the narrator's opinions as their own. In this case, your concern is moot, as the reader will probably never identify with this society in the first place. Despite any relatable behavior members of the society might demonstrate later on, most readers will be more comfortable identifying with the narrator.
Of course, this is an issue, because you DO want the readers to identify with that society. Instead of worrying about readers feeling insulted and giving up, you should really worry about readers finishing the book and feeling vindicated, not introspective. "Oh, I've done that before" seems integral to what you're trying to accomplish here, and rightly so. This is a worthy objective. What stands in the way of this objective is how comfortable it is to identify with the narrator and how uncomfortable it is to identify with the society. You have to make sure that the narrator, though victimized, is not a perfect victim, and that your society, though guilty, is not monstrous, and, above all, you need to establish this from the get-go. Doing so in the first person is a bit tricky, especially if your narrator believes that they ARE a perfect victim and that society IS monstrous, but it isn't impossible. Consider the opening lines of The Great Gatsy:
In my younger and more vulnerable years my father gave me some advice that I've been turning over in my mind ever since. "Whenever you feel like criticizing any one," he told me, " just remember that all the people in this world haven't had the advantages that you've had."
By invoking this bit of paternal wisdom, Nick, the narrator, implies that he, like his father, values having empathy for others. This fact, in tandem with his status as narrator, inclines us to identify with him. However, his saying so should raise alarm bells. Imagine, if some stranger came up to you and said "My father always told me to respect others, and I think about that every day." That's a pretty conceited thing to say to someone you've just met, but that's exactly how Nick introduces himself to the reader.
The key here is having your narrator say one thing, but betray something else in the process. If your narrator always describes their outfits in lavish detail, most readers will rightly suspect that they're a little vain, even if the narrator claims otherwise.
Another tool is dialogue. Even in a first-person narrative, all description is filtered through the narrator's biased perspective, but dialogue doesn't have to be. Try having a character say something vague, only for the narrator to assume the least charitable interpretation possible. This might signal to the reader that the narrator isn't the most objective judge of character, especially if the narrator's interpretation is actually wrong.
By all this, I don't mean to argue that your narrator must be just as evil as the society around them, but they do need enough grime that the reader doesn't wholly identify with them. The inverse is true of your society. You don't need to make them good, but if their only relatable traits are bad traits, people aren't going to identify with them.
Consider that your story has a moral
I don't think it should. The fiction author has no right to "instruct" the reader. To say otherwise is to imply that the author is somehow superior to the reader. Even if this were true, fiction deals with emotion, not logic, so any argument presented through it is bound to be manipulative. A novel arguing against abortion isn't particularly valuable. Those who already agree won't learn anything, and those who disagree will rightly write it off as propaganda. Yet, a novel that explores the topic of abortion, which depicts characters both for and against it, and makes a genuine, empathetic attempt to understand why these characters believe what they do, will undoubtedly leave the reader with something to think about. Of course, this paragraph might just be a wasted effort. I think you already understand this when you say
I want it to steer the reasonable reader to introspection rather than feeling the story is finger-pointing
If you want to inspire introspection here, you need the reader to see themselves in bad people. Yet, all people, even bad people, have both good and bad traits. If you only give your society characters bad traits, the most they'll ever be are collections of bad traits, not people.
