Don't forget that European royalty conducted a long-running experiment on the effects of inbreeding, and the result was a great deal of deformities and disabilities. A whole genetic disorder is named this way: just look up "Habsburg jaw." Lots of other experiments on human inbreeding have been run on islands like the one in the story.
Your post sounds an awful lot like, "I don't understand, therefore it is not understood."
The level of inbreeding on that island is quite scary. Although if they purged a lot of the deleterious alleles from the population, if they outbred with others, they could produce much healthier offspring and a lower likelihood of various genetic ailments.
And not the general statements that have been made in this thread, but something that would pin down a specific numeric relationship between population size in humans, and health.
EDIT: and for reference, the prevalence of Asthma may be purely due to the founder effect, not an effect of inbreeding per se. Two or three of the original settlers suffered from asthma[0].
http://onlinelibrary.wiley.com/doi/10.1111/j.1365-2222.1974....
Rougly 1.5 new deleterious mutations per diploid genome per generation. If you have closely related individuals having children, there is a greater chance that the deleterious mutations will overlap.
The claim was that the population size of Tristan da Cunha is problematic. I still haven't seen any direct evidence presented of this fact.
You linked to an article about the higher incidence of asthma on that island yourself (above), and the reason why the number of individuals is so high directly traces to the islanders' gene pool.
However, I wouldn't say the population size is unsustainable. They are more likely to be decimated by a random disease or, over generations, by a shared mutation. But the population is not nonviable. The issue is more that, to the modern humanitarian mind, they will have to deal with more ill effects due to the genetic monoculture.
> purely due to the founder effect, not an effect of inbreeding per se
Genetic bottlenecks greatly amplify the effects of any genetic abnormalities for obvious reasons. It sounds like you assume inbreeding and genetic scarcity are two different things when they're not.
Yes, the founders brought these genes into the population. That is the whole point when we're making observations based on the inbreeding which caused a higher incidence of symptoms as compared to the mainland human population - exactly because there is a high chance for recessive traits to be expressed.
If you mean inbreeding does not necessarily lead to a higher mutation rate, you are correct. But that was not implied by the comment you're quoting.