Why is the undergraph definition of Lebesgue integral so rare?

There are two problems with the undergraph definition:

• If $f: \mathbb R \to \mathbb R$ you would need to define $m(Uf)$. Note that $Uf \subset \mathbb R^2$ whereas you define $m$ as a measure on subsets of $\mathbb R$.
• Even once you have defined the product measure (the one that can measure sets in $\mathbb R^2$, the whole thing being dreadfull) you would need to prove that $Uf$ is measurable.

So, why this abstract definition? Simplicity.

Wheeden and Zygmund's book does something very clever with this. They construct Lebesgue measure on $\mathbb{R}^d$, then define the integral via the undergraph. Have a look.