Restricted maximum likelihood for mixed models

Question

We have the following mixed model: $$y=X\beta+Zu+\epsilon $$, where $u \sim \mathcal{N}(0,\tau I_d)$ and $\epsilon \sim \mathcal{N}(0,\sigma I_p)$. When using restricted maximum likelihood for mixed models (REML), we use some matrix W which has the propriety that $WX=0$, then we can just calculate the maximum likelihood of $$ Wy=WX \beta+WZu+W\epsilon=WZu+W\epsilon$$ to find $\tau$ and $\sigma$ and then we can get $\beta$ by maximum likelihood using the previous reults. But I don't understand why we don't we use another approach where we just choose W such that it "kills" the variance term Z. In other terms, we want $WZ=0$ so that we can solve by maximum likelihood: $$Wy=WX\beta+W\epsilon$$ and use maximum likelihood to find $\beta$ and $\sigma$ and then use those results to find $\tau$. I find the second approach more intuitive but from the sources I read they don't even mention the second approach so I assume there must be a good reason for that. If there is good reason for that, what would it be?