This, to me, is a general problem with a purely quantified, metrics driven approach to management, not just limited to NPS. I'm not demonizing these approaches, just saying that you have to balance humanity with data.

I like to look at this kind of thing as the one of the dark sides/dark patterns of using data for decisionmaking.

Qualitative measures are important, as is maintaining as much humanity as possible, to have a balanced and healthy culture. Being solely metrics/data driven can lead to cold, heartless, damaging culture (might be efficient or make profit, but very dehumanizing).