I was wondering about NPS quite a bit myself but never enough to research more in depth. This answer is to share what I learned about NPS.
There is research on the effectiveness of the classic 10-point-NPS question and how it could be improved.
A summary of a research paper on MeasuringU describes research and experiments especially with the scale (10-point vs. more or less and labels for the points).
It's probably best to read the text behind the link yourself, so just quoting the researchers' conclusion here:
We did find that reducing the number of scale points to 7-points
generally improved the validity of the measurement. However, contrary
to our expectations, assigning full-labels did not improve the
validity, it rather produced weaker relationships between the scales
and the validity criteria. … Our results show that different measures
such as likelihood of recommendation, satisfaction and liking are
interrelated and might be acting within causal chains.
The takeaway of the MeasuringU author is:
Similar to other studies, they found that the NPS and multiple metrics
correlated with historical growth rates and changing the number of
scale points and labels improved correlations, but not by much.
Some time ago I read (or heard) about problems with the typical NPS question ("How likely are you to recommend our service to a friend") and they resonated well with me. One problem is the point in time when this questions gets send out, typically by email. Directly after an interaction? If not, how long after. Does the customers still remember? Another problem is the question itself. It is asking about a potential future event (how likely are you) based on an interaction in the past. A better question to ask would be "Did you recommend our service to a friend" or "How often did you recommend our service to a friend", as this is easier to answer for the customer. (Unfortunately I can't find the source anymore)