The reason we care about extinction is that we probably caused them. The reason the police care about the death of a person you were arguing with is because you probably caused it.
A reason we care about extinctions that we cause is that interesting and potentially valuable species are casually discarded. We don't care about the extinction of the smallpox virus because it was an intentional process because of the incredible harm it caused us. If the Sumatran Rhino had a habit of killing everyone in any village it came across, we would be less concerned about their extinction. Instead the rhino peaceably goes about its life and we humans kill it out of spite, or to make our erections bigger, or other stupid short-sighted selfish reasons.
The point of no return for a biological species is a complicated concept. If you have one male and one female, then they can survive... but they will probably be more susceptible to the pathogenic organisms that can evolve faster than them. If you have one female of a parthogenic species, that species can survive. The low genetic diversity only impacts the likelihood of long term evolutionary survival, not their "viability" as a species.
In the case of "Lonesome George", it was only a hopeless cause because he no longer had any interest in sex. If he had lived another twenty years, we would probably have cloned him and generated females for him and thus preserved the species for repatriation to the island he came from. Should we just have made soup out of him when we realized we couldn't find any others like him?
Cheetahs show evidence of an evolutionarily recent extremely severe population bottleneck. Skin grafts can be made from one animal to any other random animal and have a decent change of success. Should we just kill off the cheetahs because there's no hope for them genetically?
The point of having a bunch of animals is that they have their own life and evolutionary trajectory to follow. If they're sickly, then selection will rapidly sort out the unhealthy versions of genes and the species will move along.
The "dealing with it" is the necessary testing part of medical science. It could be throat cancer, or a wasp stuck in her throat, or a magical fairy that doesn't like her, or a herpesvirus infection of the controlling nerve, or radiation damage from a neighbor's death ray experiments, or Bell's Palsy. Without any testing to validate the opinion, the opinion means absolutely nothing at all about what is actually going on in the medical issue.
Doctors will initially diagnose and treat for the common causes of a problem, because that will help the majority of people. The only way to know that a more complicated scenario is unfolding is to notice the standard treatment did nothing to help (or even made it worse). If the disorder is very rare, such as partial paralysis of a vocal chord, there will be no common cause to guide their treatment.
I agree, CNVs are really easy to detect if you have the read depth. I've been using the samtools pileup output to show CNVs in my study organism. However, to make the results mean anything to most people, I've got to do a few more steps of processing to get all that data in a nice visual format.
If you don't have the read depth, you lose the ability to discriminate small CNVs from noise. Large CNVs, such as for whole chromosomes, are readily observed even in datasets with minimal coverage.
Look up Tale-Nucleases.
They're still in the research phases, but they're the sort of technology needed to do targeted alterations as you suggest. The difficulty in the human case would be to get the protein into every single cell... but you might be able to get away with altering a batch of stem cells, which would then added back into the heart/etc to ameliorate specific clinical pathologies.
DDT leads to a thinning of eggshells in raptors, but not chickens/sparrows/crows/dugs/etc., via its metabolite DDE. There were LOTS of studies on this topic back before the general scientific consensus had been reached. Researchers stopped studying this topic because they lost interest in it.
Bald Eagles eat primarily fish. The particular fish eat primarily other fish and bugs. Perhaps you're thinking of Golden Eagles, which do primarily eat mammals on the size range of rabbits and prairie dogs. I don't know what an "American Eagle" is.