It depends on what do you mean by distortion. Perspective distortion is related to field of view, so a 35mm Lens being equivalent to 50mm lens on film body will show the same perspective.
For a long time i was confused by this. I thought that larger format cameras produce more correct perspective just because they had the correct actual focal length. I was wrong.
As you said "I figure that the
wide angle distortion won't change...", but if you put a 28mm FF wide angle on a APS-C camera, it's not a wide angle any more. It is 42mm equivalent, and produces the exact same perspective as a 42mm lens on FF.
You can do a simple test with two different lenses and cropping in post processing. If all other parameters are equal, you should get the same image (except for lens quality and resolution). Or you can test this by drawing 1D projections on paper trough imaginary pinhole.
Just make sure to crop the Center of image. Otherwise you'd get a shift lens equivalent

.
Another things is actual lens ideality. Obviously you can make better 35mm f4.5 lens for a medium format camera, than 18mm f3.2 for APS-C. Especially when one costs 1000$ and the other one is 100$. You need very high precision to correctly squeeze an image on those 1/2.5" P&S sensors. With larger pixels you can have larger tolerances as long as you cover the whole sensor.