First let me say that I am not a geneticist and may therefore be disregarded by you. I would however, implore you to consider my response as an educated and intelligent one regardless of my credentials in the genetics world.
There are many problems with the breeding of unpedigreed animals. First and foremost, that there are no genetic tests for many of the ailments which will be compounded with inbreeding. For example, there are no tests that would indicate females with small pelvis. Why is this important? Because a female with a small pelvis is not able to naturally birth her young. You mention the wild populations and how they breed regardless of ailment, the reality of that is that in the wild, chinchillas die young. All animals do, when in comparison with their domestic counterparts. This is because of natural selection. In the wild, without veterinary care, the problem would sort itself out without human heartache or headache. Referencing the specific ailment of small pelvis, a female in the wild unable to naturally birth would die in childbirth and her offspring would therefore not exist to reproduce such problem. Another ailment that cannot be genetically tested for would be malocclusion. And again referencing a wild population, breeding with disregard to health, those animals would often live long enough to reproduce (which is why, unfortunately, this particular ailment is still prevalent today). HOWEVER, in the wild there is nobody to purchase said offspring. In domestic chinchillas, owners of such chinchillas would be left with a chinchilla that would live a VERY short and VERY unhealthy life. Requiring frequent and unpleasant vet care and eventually dying of starvation. The wild population would also suffer in such a manner, but would relate to "if a tree falls in the woods and nobody is around, does it make a sound". There are no human caretakers to witness the excruciating life that those chinchillas live.
You mention that chinchillas did not ask to be bred for shows and I do agree, they did not. But chinchillas did not ask to live excruciating, stress-filled lives either and that's what many ailments would allow them. Because in the domestic world, chinchillas receive vet care. And while that on the surface seems like a great thing, a sick life for a chinchilla is no life at all. I can show you countless stories of the suffering of malo chinchillas, or the stories of mothers who couldn't birth. It's not pleasant either.
Now, in regards to the base question of inbreeding being problematic. In a well established, ailment-free pair, inbreeding for a generation has been shown to be beneficial. It is done by respected ranchers successfully for greater strengths. However, inbreeding essentially "doubles down" on what is already present. So if your chinchilla has bad (or good) genetics, inbreeding will bring forth those genetics stronger. In healthy animals, that's a great thing. In animals with poor genetics, it's simply going to make sicker offspring.
Aside from noting parentage and possible ailments within a line, a pedigree also shows occurrences of inbreeding/linebreeding. Without a pedigree, you do not know if your chinchilla is already inbred and therefore worsening the problem.
I agree that the nature of animals allows for random ailment to occur even in the healthiest of lines. That is obviously true, but randomly occurring here and there...and occurring with almost certainty are two entirely different things and without knowing genetic history you have no idea if that is what you are breeding towards.
My suggestion is NOT to not breed. Babies are fun, you're correct. However, my suggestion is to breed for healthy animals regardless of whether or not you intend to participate in the show system. An animal can be otherwise healthy, even if not show quality. However, to do so, requires knowledge of the genetic history to some degree.
And on a final note, in relation to your comment in regards to human life...I lead a happy life, and while not genetically perfect I enjoy watching my children live and breath and be happy. However, if my genetics were so that children of mine had even the slightest chance of having a debilitating disease that ran in mine or my spouses lineage...*I* would not have bred.
