The theory is based on the belief that the wolves select the weaker animals more likely to die anyway. I cannot doubt some of that happens at least by pure chance. But in a relatively stable system already populated by other predators that prey on elk (humans, grizzly bear, and cougar) the addition of wolves can only be additive. Although the wolves may indeed take some prey that would otherwise have died by natural means or the other predators, the other predators will make up for their losses of those animals.
The alternative is additive mortality; i.e. each wolf-caused elk kill adds to the total elk killed in a year. That seems a more appropriate model for adding wolves to a system inhabited by other predators, including humans.
So far although I have found some claims that observations support at least some compensatory mortality in some areas I haven't found research testing the hypothesis. Most of the research suggests "it depends" on things such as the predator to prey ratio and densities and the effects of other predators including human hunting. Some of the discussion suggests that human predation operates in a contrary manner, selecting the healthiest animals most likely to otherwise survive.
Clearly if the elk population in a wolf inhabited region is dropping that is evidence the wolf kills are not all compensatory. On the other hand some will be compensatory in all wolf inhabited environments through sheer chance.
Christianson discusses another, perhaps more important effect which he calls the "Risk Effect". Risk Effect
The figure summarizes his findings for 12 elk herds before and after colonization by wolves, showing a huge reduction in elk calf survival (recruitment) that cannot be predicted by direct predation and is not in line with predictions based on other factors such a snow pack. This suggests that the mere presence of the wolves as predators reduces the pregnancy rate of wolves leading to decline in the elk population. This more than offsets compensatory predation. He notes, " Recruitment, as measured by the midwinter juvenile:female ratio, was a strong determinant of elk dynamics, and declined by 35% in elk herds colonized by wolves as annual population growth shifted from increasing to decreasing....Collectively, these long-term, large-scale patterns align well with prior studies that have reported substantial decrease in elk numbers immediately after wolf recolonization, relatively weak additive effects of direct wolf predation on elk survival, and decreased reproduction and recruitment with exposure to predation risk from wolves."
Mech makes an interesting point that this principle can also affect the wolf population. Mech He also notes that higher wolf mortality can positively affect pup production and reduce the wolf mortality from other wolves. This leads to the need to cause relatively large (25% to 50%) reductions in the wolf population through hunting or other means to render it stable or reducing.