Is this a valid derivation of FWL?
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 $\boldsymbol{\hat\gamma}$.