I knew about this nonprofit via a cat colony caretaker, @poetssquarecats (I follow her on Tiktok and she's amazing) that talks about the struggles of feral cats and uncontrolled cat colonies. Yes, of course animals in the wild also suffer from diseases, parasites, and injuries. But these animals aren't in the wild. They're far more constrained than in the wild, with a lot more access to food, and with a lot more dangers. One of the problems is that, in those conditions, the number of cats can and does explode (female cats can become pregnant at 4-6 months old already, and they have on average 16 kittens a year in multiple litters).
And yes, of course getting rid of them is going to have unwanted consequences, but having them has other unwanted consequences. For example, as the article mentions, birdlife is severely affected, which can also have other unwanted consequences. I think that in that case it's better to have the cause under control: either way it's going to have unwanted consequences, but at least when you control it you know that it's happening.