Why is uncorrelated(exogeneity) "good enough" for identification in regression?

Question

Let the model $y=x+x^2-1$ be exactly correct, where $x\sim N(0,1)$, then $x^2\sim \chi^2_{(1)}$. Say we want to estimate the model $y=\beta x-1+\epsilon$ by least squares. Let $(y_i,x_i)_{i=1}^n\overset{iid}{\sim}$. So we should have an omitted variable bias. Solving for $\beta$, you get $$\overset{\wedge}{\beta}\rightarrow\frac{cov(x,y)+E(x)}{var(x)}=\frac{cov(x,x+x^2)+E(x)}{var(x)}=\beta+\frac{cov(x,x^2)}{var(x)}+\frac{E(x)}{var(x)}=\beta$$ as $cov(x,x^2)=0$, and $E(x)=0$. So it seems like we identified $x$ correctly by $\beta$, but the marginal effect of $x$ is $\frac{dy}{dx}=\beta+2x$, so we haven't identified the effect of $x$. So why don't we need the error term and the regressors to be independent, instead of just uncorrelated? This question is really a precursor for why in GMM, specifically IV, we need our instruments to only be uncorrelated with the error term, and not independent of it, as without that, we don't identify the complete effect of $x$. It seems like especially for non-linear regressions, we would need independence to identify the effect of our independent variable.