It comes down to this: which do you want to be simpler to deal with?
- Functions which have integrals of infinite magnitude.
- Integration of vector valued functions.
Traditionally, infinite integrals are seen as a more immediate obstacle (${\mathbb R}$ is an infinite measure space so this difficulty shows up quite quickly). Such integrals probably seem more obviously relevant to students at first. So texts usually develop integration with (1) in mind from the beginning.
To me, integration of vector valued functions is a lot more natural, and playing with the extended reals seems like little more than convenient notational trickery. But this is hindsight. I'm quite sure that when I first learned integration I would have been much more concerned with problems caused by infinity.
