For Proba-3 the goal is to have the two satellites more than 100m apart. If you want to do that with a stick, your stick has to be longer than the ISS. That should tell you a thing or two about the complexity and cost of building and launching that stick.
I do have to admit I'm not exactly sure what the advantages are of having the disk further away from the telescope. I suspect it's to do with the interaction between the light and the edge of the disk, but I'm not sure.
https://www.jpl.nasa.gov/news/starshade-would-take-formation...
Edit: Just to be clear on status of this, Starshade is still in early technology demonstration phase that they can actually build the shade and do the formation maintenance. This is not in full build or slated to launch any time soon.
EDIT: maybe not so counterintuitive after all: if you scale everything up, you get a higher fidelity sensor and more signal to noise. Same reason telescopes want to be big: to collect more light especially from dim signals. Distance of the occulter then reduces perspective distortion that would eclipse the inner parts of the sensor more than the outer ones. Just my speculation though.