My impression is that breeding from diverse backgrounds gets you hardiness-- mutts are less likely to have the specific genetic ailments you get from purebreds.

Yes, but I think this works on a different scale. Purebred domestic animals are usually heavily inbred, precisely to push a particular trait to new heights. In the standard textbook manner this makes the chances of the animal getting multiple copies of some recessive gene skyrocket, thus the fragility.

The human equivalent is marrying your cousins (inbred human populations exist, they usually don't look too good) which is different (scale) than marrying someone from a large enough gene pool (e.g. like all Europeans).