Some utilitarians say or assume that utility function has to be a linear combination of its atomic final values. It is not. Surely you can try to decompose a non-linear function into a linear combination of other non-linear functions, but this is an unrecognizable steelman of what people say when asking to shut up and multiply.
Proof: the existence of https://shipyourenemiesglitter.com/ and the fact that people actually use that.
Imagine that someone’s wellbeing has a negative coefficient in your linear utility function. Would you ship them glitter? Sure. But if you want them to suffer as much as possible (which is true if it’s a negative-valued member of the utility function), you can do much better than that. Do they pirate music and games, do they possess drugs, do they have anything embarrassing in their life? Most likely yes. You can use that to legally and with little effort completely destroy someone’s life. If you chose to ship glitter instead, their wellbeing cannot be a negative value for you.
OK, what if it’s zero or positive? Then why do you ship them glitter in the first place? Maybe you care about them, but something in their behavior is so bad that you want to teach them a lesson. No, this doesn’t work – anonymously shipping glitter, possibly even with a generic message, is obviously and vastly inefficient way to improve someone’s behavior. If you ship glitter to someone, you’re almost certainly want them to be mildly frustrated for the sake of being mildly frustrated.
Let’s go deeper! What if this person is useful to you, and you want to make their life as uncomfortable as possible, but in a way that doesn’t disrupt your getting utility from their other actions rather than just existing? Still doesn’t work – this is far from the most efficient non-life-destroying method of annoying someone.
What fits this behavior perfectly is a model where the utility of an annoying person is a non-linear function, whose maximum is at “mildly frustrated, but pretty OK in broad terms”. If they get terminal cancer, you’ll feel bad for them. If they get a huge raise, you’ll feel angry and wronged by the just world. But if they still have their job, house, and family, but keep finding glitter everywhere, you’ll feel the sweet sweet schadenfreude, which is perfectly explained by non-linear utility, and not at all explained by linear utility.