So it's important to understand why humans believe in metrics. Humans by default are multi-objective. We don't care about just one or even a few things, we care about a lot of things, all at once, with nobody agreeing on what they are. Identifying the "true direction" we want to go as a collective is impossible. So we pick a good enough direction so everyone can be somewhat aligned. To make it possible to communicate this vision, we compress the world down to few dimensions even though we know that it is absurd. This works fine for humans because our "distributed system" is composed of beings with enough intelligence to notice/alert/convince the rest of the system whenever we are veering off course.
Whenever we program AI with these metrics, we forget that we can't direct them we way we direct our sentient brethrens. Any simple metrics are necessarily doomed to fail because it doesn't have this self aligning property. So ultimately it is the programmer's responsibly to go and realign the system back with human values. If necessary, with some social pressure (hence this article). It doesn't seem like there will be any other way.
This is such an odd statement. I think the article makes a good case that in this context, metrics serve as something like the opposite of a tool - a trap, something a person or group uses that takes them something other than where they intended. Now approximation, generally, is something like a tool but it can be used well or badly.
Now, a bureaucrat or someone who want to "just get stuff done" may use a metric to get people moving without caring exactly where. But that's still with a human context, not an AI. In an AI context, a metric can run a plane into the ground or more or less anything.
IE, it's only some of the contexts (racist AI) where metrics have a social content.
The point about corner cases is reasonable but it's easy to by analogy that human correction can't keep with the problems metric yield for complex spaces; just consider how a ten dimensional cube has 2^10 literal corners and AI more or less operate in spaces with many more dimensions (of course not all "corner cases" are literal corners but there's a similar problem of intersecting "areas of concern" or whatever one would want to call them).