$Y = X\hat{\beta} + Z\hat{\gamma} + e\\ (X'X)^{-1}X'Y = (X'X)^{-1}X'X\hat{\beta} + (X'X)^{-1}X'Z\hat{\gamma} + (X'X)^{-1}X'e\\ \hat{\beta}_{short} = \hat{\beta} + \hat{\delta}\hat{\gamma} $
where $\hat{\beta}_{short}$ is the OLS estimator of the coefficient on $X$ from the misspecified model that omits $Z$, and $\hat{\delta}$ is from the regression $Z = X\hat{\delta} + \eta$.
This seems infinitely more simple than FWL proofs I've seen in econometrics textbooks. So what is this missing or getting wrong?