No inbreed does not cause a problem but it does make it more visible. As an example tay sacs in humans. Its a recessive gene that improperly folds proteins leading to a build up of toxins and plaques that cause death. Its common for people to be genetically tested so that they do not produce a child with this disorder. As a recessive gene you can carry it and not be affected. If you produce a child with someone with the gene you get a manifestation of the disorder. Inbreeding did not cause the disease but it did make it manifest. Look at the highly unique disorders (psychological and physical) that affect many in the Amish community. This is the result of a relatively small gene pool resulting in recessive traits expressing. So, line breeding does not cause a disorder (unless we start talking about spontaneous mutations that are also recessive), it does cause the manifestation of traits.
In horses, someone mentioned morgans going back to one stallion (Justin Morgans horse). While they are all line bred they are also bred for a "cresty neck" that cresty neck which could be an offshoot of inbreeding/line breeding is also associated with metabolic disorder which results in laminitis. One could say that metabolic disorder in Morgans is a result of them being linebred for a specific trait.
Horses in the wild do inbreed but probably not to the same extent or intensity as captive horses do. When wild horse do inbreed/linebreed in the wild they are acted on by often cruel natural forces that result in the death of those unsuitable. The number of HYPP horses and Frame/Frame breedings that occur or are asked about on this board alone indicate that these deletrious effects are not often selected against and are sometimes selected for.
I completely agree that people should research and know the genetics of their horses. Those that do that are in the minority. This is in part because most people feel their horse is outstanding. If people did not think their horse was breeding quality they would not breed it. Most people feel their horse has a great personality, flashy color, its out of XXX stallion or it would be a good mother etc. Not many say, I want to breed it and Y is in the breed so I should test for that.