Actually, in relation to cats and dogs, chinchillas haven't been around that long as pets at all they were first brought to the US from the wild in the early 1920s, and it takes generations to go from wild to domesticated and really only being in the more mainstream pet world for 40 years or so. In reality most small pets you get are considered exotic, they are not native to the area you live in, so hamsters, gerbils, guinea pigs, most birds, reptiles, etc are all exotics unless you live in an area where they are native to. They do teach about chinchillas, and other exotics in vet school, but briefly. It's like a brain surgeon for example, they are a doctor but require extra training in a specific field, doctors are all taught about the brain in medical school but you can't expect your family doctor to be able to preform brain surgery on you. Exotics are a specialty that vets go into after the basic vet school, and requires more specialized schooling. I guess you can also look at it in terms of, you wouldn't bring a horse to an inner city companion animal vet clinic, you'd take it to a vet that specializes in livestock, a farm vet. If you stop to think about it, a general vet has to know the anatomy and behavior of hundreds of animals, you are better off having someone that has devoted their career to just a few (which is one reason cat only vets or dog only vets are becoming popular) rather then a general vet that mostly sees cats and dogs and may never have even seen a chinchilla in real life. There is more money in specializing for a vet, but not guaranteed work, so most stick to mostly cats and dogs.
Also chinchillas aren't really that common yet, in the last 20 years or so their popularity has increased a lot, but still how often do you really see chinchilla breeders around? In comparison to how often do you see people breeding cats and dogs? Most people in the US have had a cat and/or dog sometime in their life, or know someone who has.