real analysis - Why is Lebesgue integration taught using positive and negative parts of functions?

Well, it is not true that Riemann integral and series avoid the distinction altogether.

If you want to define improper Riemann integrals, you can follow two ways. Say you want to define $int_a^b f(x) dx$, where $a, b$ are not necessarily finite and $f$ need not be bounded.

Either you split $f$ into the positive and negative part, or you define it as a limit of the truncated functions on a truncated domain, something like

$$lim_{t to +infty} lim_{s to a^+} lim_{r to b^-} int_s^r max { min { f(x), t }, -t } dx $$
But then the result, for functions not in $L^1$, depends on the way you choose to go to the limit.

Exactly the same happens for series: for those which are not absolutely convergent, the result depends on the order of summation.

The trick to consider $f_{+}$ and $f_{-}$ allows one to consider improper integrals, which may as well be infinite, and to declare that the integral of $f$ does not make sense in those cases where the order of the limits is relevant.

Of course you know all this, but I post it as an answer since it would be too long for a comment, so that you can comment to explain why this reason is not enough for you.