Not really true in practice. FE (and UWA) distorts more the closer the subject is to the lens and that includes straight horizontal lines at the centre of the frame. What's more, to minimise the distortion, the camera has got to be dead centre perpendicular to the horizontal lines.
If the straight line is the horizon very far away at the centre axis, the camera is for all intents and purposes pointing perpendicularly to the horizon... hence the straight line at the horizon.
If the subject with horizontal lines is in the mid-distance or close to the camera, it could suffer distortion even if it is at the centre axis as most of the time it is not practical to be perpendicular to the subject because of space or compositional constraints. Even if the distortion at the centre axis is minimised, the rest of the subject slightly off-centre would still distort when shot with a FE, most of the time in a visually unesthetic manner. This would be where a UWA would be better off. Example...
This is not a keeper for me.