What's the use of scaling here?

Consider the following exercise:

Suppose that a mathematical modelling is given by the following equations: \begin{align} \dfrac{\partial u}{\partial x} + \dfrac{\partial w}{\partial z} = 0, \tag{1}\\ \end{align} \begin{align} \rho\bigg(u\dfrac{\partial u}{\partial x} + w\dfrac{\partial u}{\partial z}\bigg) = \eta\dfrac{\partial^2u}{\partial z^2}.\tag{2} \end{align} We apply the scalings $$\overline{u} = \dfrac{u}{U}, \overline{w} = \dfrac{w}{W}, \overline{x}= \dfrac{x}{X}, \overline{z} = \dfrac{z}{Z}$$ with $U$ given and $X, Z,$ and $W$ to be chosen later on.

Show that in $(1)$ and $(2)$ the number of parameters reduces if the following two conditions are satisfied:$$\dfrac{XW}{ZU}=1,\dfrac{X\eta}{Z^2U\rho} = 1.$$

I think I've managed to show that this holds. Using the scaling we get the equation $$\rho\bigg(U\overline{u}\dfrac{U\partial\overline{u}}{\partial \overline{x}} + W\overline{u}\dfrac{U\partial\overline{u}}{Z\partial\overline{z}}\bigg) = \eta\dfrac{U\partial^2\overline{u}}{Z^2\partial\overline{z}^2}$$ If we fast forward through some calculations we get: $$\overline{u}\dfrac{\partial\overline{u}}{\partial\overline{x}} + \overline{w}\dfrac{XW\partial\overline{u}}{ZU\partial\overline{z}} = \dfrac{X\eta\partial^2\overline{u}}{Z^2U\rho\partial\overline{z}^2}$$ which means that if $$\dfrac{XW}{ZU}=1,\dfrac{X\eta}{Z^2U\rho} = 1.$$ we've effectively removed the parameters $\rho$ and $\eta$.

Question: I don't really think I understand the concept of scaling applied to these equations though. True, we have reduced the dimensionality of the set of parameters, but aren't there easier ways to do so? What if we just divided both sides by $\rho$. Would that mean that we would have reduced the number of parameters by one as well?

It's a bad example having $\rho$ on the LHS here as this parameter can be gotten ridd of without scaling as you point out. However, in any case you would still need scaling to get ridd of $\eta$.

The main point of scaling is that instead of having to solve a PDE for every choice of parameters (here $\rho,\eta$) independently we solve it for one set of parameters (say $\eta=\rho=1$) and use scaling to get it for other values we might be interested in. This makes it a very useful thing to know about.

Another thing missing in your approach is to apply the scaling to the first equation also:

$$\overline{u}_{\overline{x}}\frac{U}{X} + \overline{w}_{\overline{z}}\frac{W}{Z} = 0 \implies \overline{u}_{\overline{x}} + \overline{w}_{\overline{z}}\frac{WX}{ZU} = 0$$

which we see leaves it on the same form as before since $\frac{WX}{ZU}=1$.