I studied the Lebesgue integral from the first time in a book called "Analyse réelle et introduction aux méthodes variationnelles", writen by a couple École Polytechnique teachers, which defines the Lebesgue integral by means of Levi sequences. From now on, I shall call this method "method 1".
However, after that I read a little Stein's and Shakarchi's wonderful "Real Analysis", which defines the Lebesgue integral in the (I think) usual way: using simple functions. From now on, this will be "method 2".
It seems to me that method 1 is much more cumbersome, in that it makes proofs harder, the intuition not as clear and is overall worse. I know that, using this method, it becomes clear that if $f$ is Riemann-integrable, then it is Lebesgue-integrable. However, the proof of this fact using the second method is not so hard to compensate the huge list of theorems that become easier using the second method.
The people that wrote this book (and a couple other mathematicians which wrote books utilizing this definition, including Riesz) are surely not dumb. They obviously have a reason to utilize this definition. I just don't get it. Why?