It is much more complicated than that. There are 85mm prime lenses I'm aware of that exhibit some degree of barrel distortion, and barrel distortion is almost a given at the wide end of a zoom lens no matter how long that wide end is. On the other hand, at the "long" end of the Tokina 11-16mm f/2.8, you would have to be shooting brick walls to notice any barrel distortion, and even then it's only at the very edges of the frame (on a crop-sensor camera, of course). And the very same lens may exhibit different geometric distortions at the same focal length but at different focus distances. Modern lenses are a complex interplay between a lot of different lens elements that often change their spatial relationships as the lens is adjusted (focused or zoomed).

There are several prime lenses under 20mm that come close to being properly rectilinear. Sigma's 14mm lens was great in that regard, but a lousy lens otherwise, unfortunately (it was terminally soft anywhere but at the center of the image, had poor contrast, flare everywhere, etc.). Nikon and Canon both offer 14mm lenses that have slight and easily correctable barrel distortion (slight being relative in the extreme wide-angle world).

The good news is that geometric distortions are fairly easy to correct in post. (And recent Nikons will do it in-camera; I don't know about other brands.)

The bad news is that even if gross geometric distortions are corrected, that does nothing about the foreshortening distortion that makes spheres appear like ellipsoids and gives people at the edge of the frame funny-shaped heads.

That's not got anything to do with the rectilinearity (the absence of barrel and pincushion distortion) of the lens, but with perspective. In order for things to look right, your point of view (when looking at the picture) needs to be about the same as the camera's point of view was when the picture was taken. (Give it a try -- if you have a picture that has perspective distortion, just get your face really, really close to the print.)