Is this a valid derivation of FWL? $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?
The result is not the Frisch–Waugh–Lovell Theorem. Rather, the result shows the extent of Omitted variable bias.

The Frisch-Waugh-Lovell Theorem states that the residuals of the regression of $\boldsymbol y$ on $\boldsymbol Z$, regressed on the residuals of the regression of $\boldsymbol X$ on $\boldsymbol Z$, gives $\boldsymbol{{\hat\beta}}$. To put it another way, let $\boldsymbol{M}$ be the residual maker. Then the regression of $\boldsymbol{M_Z y}$ on $\boldsymbol{M_Z X}$ gives $\boldsymbol{{\hat\beta}}$. The equation is:
$$\boldsymbol{\hat\beta} = \left(\boldsymbol{X}'\boldsymbol{M_Z}'\boldsymbol{M_Z X}\right)^{-1}\boldsymbol X'\boldsymbol{M_Z}' \boldsymbol{M_Z y}$$
Note that the residual maker matrix is symmetric and idempotent.

An analogous result holds for $\boldsymbol{\hat\gamma}$.