The example in this video was inspired by the Zo hadden we het niet bedoeld book by Jesse Frederik. The book is about the recent "child benefits" scandal in the Netherlands. There's an anekdote about how the tax office dealt with parents who, desperately, needed help to understand what went wrong. Some of these parents had to wait for months before getting a reply. One supposed reason for the long waiting time, was the metric explained in this video.
This effect has a name: Goodhart's law. It states that:
"When a measure becomes a target, it ceases to be a good measure"
If you search the web for it, you'll come across many examples.