Firstly, a lot of endangered species are a bit particular about how and when they breed. Secondly, breeding an endangered species with a limited gene pool requires very careful management to minimise inbreeding depression.

How would you co-ordinate multiple private breeders, who have their own incentives, to save a species?

Thirdly, look at tropical fish and, well, dogs. People breeding them tend to favour particular traits that lead to deformed fish that would never survive in the wild, or dogs that can't give birth naturally anymore. How do you ensure that the private breeders are breeding for genetic robustness, not a particular trait that's appealing to buyers?

Lastly, and most importantly - what happens when you legalise ownership of endangered species, and captive breeding can't keep up with demand? If you guessed "poaching", you guessed right.

Then there's also the fact that even if captive breeding could meet all market demand, the fact that now everyone in the lizard collecting community can have a jewelled gecko can drive certain collectors to want to own a "real" one, a wild one, to stand out from the people who have captive bred ones.

This drive from aficionados to have the "authentic" product can be seen, for an example I can speak on, in venison exports from New Zealand to Germany.

Germany loves venison, it's hard to come by in Europe, we can farm efficiently in NZ, and so we farm deer, butcher them, and export the meat to Germany. (NB: when I'm talking venison here, I'm referring to the meat of the red deer, Cervus elaphus. NZ has many other introduced species of deer, but reds are the most widespread and numerous, and also the one we have learned to farm the best, thanks to the market in Asia for deer velvet[0]).

But a lot of German consumers want "real" venison, which means wild venison.

So we market farmed venison as "cervena" to differentiate it from wild vension, and then use helicopters to harvest thousands of tonnes of wild venison from public land (called Wild Animal Recovery Operations, or WARO for short, actually a really great way of keeping deer numbers below ecosystem degrading numbers).

The price premium it attracts makes the helicopter flight hours needed to harvest wild venison actually economical. [1], [2], and a perspective common of recreational hunters who despise WARO operations because no-one wants to walk into the mountains for two days, and then watch a helicopter shoot everything moving in the valley. [3]

But the thing is, wild venison is always worse eating than farmed venison. It's far leaner (and venison is lean enough already, a larding needle is essential for cooking venison roasts[4]), tougher, it's gamier, it's just a poorer eating experience.

Yet because German consumers consider it more "authentic", it attracts a premium.

This is most likely what would happen to endangered species where you could buy a captive bred one, or a wild one.

[0]: https://en.wikipedia.org/wiki/Velvet_antler

[1]: https://www.youtube.com/watch?v=Ptfr-LyXo8s

[2]: https://www.youtube.com/watch?v=bKWvsS-z9PU

[3]: https://www.youtube.com/watch?v=LmvZ3y3RMXs

[4]: https://www.meilleurduchef.com/en/recipe/to-lard-a-piece-of-...