Can a fully tax-funded public school use music licensed under CC-BY-NC in a video to promote its educational programs?
[mandatory disclaimer: I am not a lawyer]
Let's start with the easy part - any usage of a CC-BY-NC piece must include proper attribution, so the video would have to include such attribution (presumably in the end credits?)
The tricky part is interpreting the Non-Commercial part of the license. First, as the CC wiki states:
NonCommercial turns on the use, not the identity of the reuser.
So the fact that the school is fully funded by taxes isn't the key point of this discussion. How this funding works probably is.
If the school just gets a flat sum no matter how it uses it, and has X open positions on this program, first-come-first-served, there's no monetary gain from producing such a video, so this usage would probably be allowed.
On the other hand, if the school's funding depends on the number of students that enroll to this program, such a video could be interpreted as an attempt to create a monetary gain (i.e., "commercial", under the license's terms), and would such be forbidden.