I am rather confused by what's written in the green box.
If $\frac{\partial A}{\partial x}$ was not $<<1$ would this mean that the velocity now has a vertical component and is hence not one-dimensional?
E.g in the picture below the cross section at $x$ has velocities on it acting at different directions.
Yes indeed, if the cross section arrea would change rapidly, then their would be a non-zero velocitycomponent in the vertical direction too. In that case, the Navier-Stokes equation would be to hard to solve.