Europe and North America had a very situation in the 20. century: massive keeping of native birds, which was threatening wild populations. For many poor people, a bird in a cage was a rare entertainment he could afford.
It was solved by the radical ban of keeping all native birds. Bird keeping community remains lively, and shifted to exotic singing canaries, small parakeets and estrildine finches which are easy to breed in cages in many varieties and races – actually domesticated. Bird keepers became bird breeders.
The biggest achievement was actually raiding bird markets and trade centers. This is the part of the bird trade chain which is most easy to control. Keeping native birds was never 100% eradicated, one still sometimes hears a native goldfinch or greenfinch from inside somebody's house, but it is sufficiently small not to depress wild populations.
Breeding native birds in cages cannot realistically solve the problem. Simply, it provokes massive catching of wild birds later sold as cage-bred ones, which are practically difficult to separate. And majority of attractive native birds are too sensitive to breed in cages on an industrial scale. That is why they should be replaced by easily bred exotic species like canaries, budgerigars etc.