Wanted to ad, just having a rescue chinchilla says something. While not currently to the extent of dogs or cats, no, "rescue" chinchillas ARE on the extreme rise. Period. Maybe it doesn't show as much...but I see cases of dogs being surrendered...mostly one at a time...then I see chinchillas being surrendered and it's rarely one at a time.
Point being, there are already SO many chinchillas in rescue...and not enough rescues to go around. Furthermore, chinchillas (in general) have a longer life expectancy than dogs or cats. Even the best quality chinchillas get passed around a few times in their lifespan on average because of this. When I breed a chinchilla, and sell it, I always tell the buyer they have the option of bringing it back to me. Period.
A breeder of any animal should have a goal in mind when breeding. The front of that goal should be healthy chinchillas and (IMO) good temperament. While a lot of breeders don't consider this an issue, when any chin winds up in rescue temperament IS an issue.
SO, health is the top consideration...if you have no known background you cannot tell if there are any defects in the lineage. You can breed a perfectly healthy one year old...if it's a female by age two you may already have two litters and 2-6 (or more) kits. At age two you get a weepy eye...teeth grinding...vet trip. Malo...all her kits are now at LEAST carriers.
Next consideration would be quality. You can look at a nice dark standard, big and blocky, perfect speciment. No known history. You breed it with the hopes of getting nice big blocky dark standards. All you get are narrow, runts with poor, and light, veiling coverage. Why? Well because your chin is a fluke, his parents were both poorly bred, poorly veiled animals that came from long lines of the same. So that's the quality aspect, as an example.
Finally, color. You're hoping to breed nice grays as that is your GOAL (keep this GOAL thing in mind). You pair your standard up with another standard. Both appear to be decent quality standards. You get a bunch of muddy bellied, off color babies. WHY? Because somewhere back in the lineage was a poorly colored ebony.
Another thing to consider is say a new mutation pops up in your pairings. You get something unique, something worth working with. For arguements sake, you pair up your gray chin with another gray chin (to simplify) and get (viola) a black velvet. This is the most stunning black velvet the judges have ever seen. It takes GSC easily...you now have a worthless mutation that you cannot figure out how you got because you do not know anything about the parents.
There's just no purpose in breeding a chinchilla of unknown heritage. The outcome would be more pet chins which may or may not be healthy and, if well tempered enough, you can easily place in some pet homes. Fine, all well and good, until...wait...there are no more homes...or your chin is bounced around to 7 different homes in two years. Who benefits from that? Surely not the rescues
Have a goal in mind. If you want to breed, why not spend the extra penny for a nice animal and go to some shows...get educated...do things right.