The British Empire drew some arbitrary lines on the map to divide up the area of the fallen Ottoman Empire, and mashed together disparate, rivalrous groups.

I agree with you, but be fair Iraq as a concept is really old, so it's not like demarcating the area we call Iraq as one administrative unit is a new idea. BTW the Iraqi state is only as functional as it is because opposition to it coalesced around ISIS and politically and military burned out/was crushed, allowing it to maintain a measure of monopoly on violence. It's still very dysfunctional in a day to day level, see: https://en.wikipedia.org/wiki/Democracy_in_Iraq.

so why would they fight for a nation-building project that the US tried to impose at gunpoint?

That too, but also intentional and unintentional Western sabotage of these nation-building project to make the results more pliable to Western interests results in shitshows like the US-backed Afghan government that folded within months of US withdrawal. It's not like the US couldn't undertake a successful nation-building project if it really wanted to; it just doesn't.