I think this is a perfectly humane solution. Especially considering it in the context of their cultural embedding, in which animal welfare in general are not the greatest. And also considering that the alternative is that feral cats starve to death.
Given that the goal was for no more cats to be born (so that the cat population eventually drops to 0) and this was apparently accomplished, it seems like a fair term.
"Driving into extinction" is an awfully loaded and hyperbolic phrase for "neutering a single colony of feral domestic cats." It makes it sound like an atrocity. Maybe that's not what you intended.
Preventing unwanted and uncared-for kittens is a good thing. It's a major goal of animal welfare groups all over the world.