The Motivation of Pre-measure for Construction of Measures

Note that even when starting with a premeasure $\mu_0$, we still construct an outer measure $\mu^*$ from it, and restrict that to a $\sigma$-algebra contained in the $\mu^*$-measurable sets. So, Caratheodory's extension theorem is a more refined statement of Caratheodory's "outer-measure theorem" regarding what happens when we impose additional hypotheses. So these aren't really two different methods for constructing measures. One just gives us more refined conclusions than the other.

The benefit of the premeasure approach is that first, the algebra we start with will be $\mu^*$-measurable and second, that $\mu^*$ agrees with $\mu_0$ on the sets of the algebra (hence it's called Caratheodory's extension theorem). We do this not just for product measures, but also for Lebesgue measure on $\Bbb{R}^n$.