Or wild-caught fish for that matter.

The problem comes with the economic pressures to cut corners and increase profits, and this comes inevitably when unregulated competition gangs up on a renewable but finite natural resource. Any natural system can be harvested indefinitely for surplus production if it is done wisely, and if the resource is properly managed. The very fact your fish farm feels the need to publicize their good management practices tells me the problem is out there and they are getting heat for it. And who monitors and regulates Chilean fish farms? That's why its so cheap we can afford to put it on jets and fly it to US markets.

If we make it a point to limit fish catches by size, place and season, to wait till after they breed, and to relax harvesting when stocks are threatened, we can have healthy fresh seafood forever, but it will cost us more. But this means government marine science and regulation (boo, hiss!).
If we want lots of cheap fish we can only do it for a while, until the fishery collapses. The market only wants as much fish as cheaply as possible. Sound management practices can only come from human intervention, not from the market.

Private fishing boat owner-operators recognize their long-term livelihood depends on preserving the resource and sacrificing short term profits for long term viability. OTOH, absentee corporate ownership, desperate to pay off their initial investment as quickly as possible and continue increasing profits, does not give a shit. When the fish run out they will invest in something else. They would go fishing with dynamite if they thought they could make more money at it. That's how they fished for abalone in California--drag trawls across the bottom and pick out those delicious shellfish from everything else that came up in the nets. Meanwhile, the kelp beds that supported the abalone to begin with were devastated, it would take them years to recover and become productive again. And not just abalone suffered, everything else that lived in the kelp died too. They call it "by-catch", as if it was just worthless trash.

The key to fisheries management is studying the life cycle of each species and harvesting it only when it is least vulnerable, protecting the environment and by-catch and limiting catches to what that population can tolerate. But the response of those with no family or traditional stake in the fishery, is to just get bigger boats, go farther out, and stay there till the holds are filled. And when the fish disappear, switch to another species, or move to another part of the sea.