potential - When to use Poisson's equation in electrostatics

When does when elect to use Laplace's equation when dealing with charge distributions. For example, if I had a metallic sphere of radius $R$ and charge $Q$, then

$$\mathbf E = \begin{cases} 0, & \text{for } r < R \\ \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}, & \text{for } r \ge R \end{cases}$$

The potential at all regions of space can be calculated in a straight-forward way then.

For $r \ge R$,

$$V = - \frac{Q}{4 \pi \epsilon_0}\int_{\infty}^r 1/r^2 \ dr$$

And for $r < R$

$$\implies V = 0$$

However, I could've just as easily tried to work it out by saying for $r > R$ and $r:

$$\nabla ^2 V = 0$$

As all charge resides on the surface, and then say that

$$V(r,\theta) = \sum_{l=0}^\infty \left(A_l r^l + \frac{B_l}{r^{l+1}}\right) P_l (\cos{\theta})$$

Wouldn't this be an equally valid way to derive the field? If so, when is it better to derive things this way rather than the way I did with Gauss's Law?