Plant based proteins are more sustainable. With animal proteins, people don't think about how inefficient it is to have to feed those animals before they end up on your plate. If you want to liken it to computer science, its a O^2 kind of inefficiency.
The only sustainable diet is one where the total has a environment zero or even positive effect. Honey would be one such animal product, since the work that the bees does has a net positive effect on the environment. Meat from over populations would also have a net positive effect, and is part of the bigger picture when farms use fertilizers and cause eutrophication.
In order for a biological system to remain diverse and productive indefinitely (the definition of sustainability), we need to look at the whole picture. Animal proteins is part of that chain, but a sustainable diet would look very different from a regular meat rich diet. The species, the amount, and also the plants would all have to be very different.
that doesn't mean its sustainable. And what would you feed these fish farms?
Cows and pigs need lots of water and lots of food (soy, corn, etc). Considering all of the required space for the animals, and the space to grow the food for the animals, and all of the required gallons of water for the animals and to grow their food, and all of the antibiotics the animals are given, then no, its not sustainable or scaleable.
I don't know where you got the idea that they are keeping up with productivity improvements, or if you are simply making assumptions. The meat is the lowest quality its ever been and line speeds are too fast to be properly inspected. Animals have to be given so much antibiotics and medications.