I think the impulse to "lift" a function on $G/H$ to $G$, and define the integral on $G/H$ as the integral on $G$ of that lift, is entirely reasonable... but doesn't quite work out, no matter how one struggles. A prescription for seeing the problem in various (futile) attempts resides in the key point for the successful characterization: letting $\alpha$ be the averaging map $(\alpha f)(x)=\int_H f(hx)\,dh$, with the sided-ness of the measure easily determined... the requirement is exactly as suggested in the question:
$$
\int_G f(g)\,dg \;=\; \int_{G/H} (\alpha f)(\dot{g})\,d\dot{g}
$$
where the $\dot{g}$ is the coordinate on the quotient. To make this be an unequivocal characterization of the integral/measure on the quotient, one must check that the averaging map is a surjection of $C^o_c(G)$ to $C^o_c(G/H)$, which is indeed so, in general.

Then, to return to the question of making a "section" of the averaging map... if the condition on "modular functions" is met, and if continuity is not necessary, then this is certainly possible. But this might be a less than optimal approach, since we might want to characterize measures/integrals via Riesz' theorem about functionals on $C^o_c$.

NB, as @KConrad suggests, the modular-function condition is *necessary*, as one fills out the alleged proof that such a formula gives an invariant measure. The popular cases, such as $\mathbb R \to \mathbb R/\mathbb Z$ with section given by multiplying by characteristic function of an interval, so subliminally assume the modular-function condition that we may not have realized.