"There's only a couple of spots that can mutate without the mitochondrium conking it, and they lead to very rare, and ultimately lethal diseases..."
You might not have noticed but there are some stretches of mtDNA which are able to show variation. This is what is used in mtDNA typing as sold to genealogists. Although there are differences between different inheritance lines the mitochondria within individuals show a remarkable absence of variation, even in these areas where variation is able to take place.
What the paper being discussed shows is that the eucaryotic host, for want of a better word, seems to exert some control over the multiplication of mitochondria when an artificial process* creates a mixture. This provides at least a partial explanation of how the variation might be reduced.
Such a mechanism is unlikely to exist for the sole purpose of defeating IVF (although Creationists might disagree with that) nor for aiding businesses selling services to genealogists so it's likely that it exists as a result of an evolutionary process.
A functional mutation in a mitochondrion might not necessarily be fatal to the mitochondrion itself - after all, as you pointed out, the mitochondrion is supported by the host cell and, of course, by its fellows. However, if such functional mutants were able to spread they would damage the overall performance of the host organism. Some sort of mechanism for preventing such a spread would provide a selective advantage for the host and the research presented here suggests that genes of the eucaryotic nucleus are indeed involved - just what one would expect from a selective advantage on the host.
The bulk of the Nature paper is paywalled but the more general summary here, http://www.npr.org/sections/health-shots/2017/01/01/507244429/unexpected-risks-found-in-editing-genes-to-prevent-inherited-disorders also linked in the parent El Reg article, would be worth a read. The rates of mitochondrial mutation mentioned there might surprise you.
*Used to attempt to overcome the very mitochondrial genetic diseases you mentioned.