Sorry to hear about NHS at your daughter’s school, TheGFG. I think that admissions committee members have grown knowledgeable about the frankly weird-o NHS policies at some schools, and don’t regard its absence as any sign of a problem.
I also understand that if it would be meaningful to your daughter (and it is clear that she deserves the recognition), it would be nice for her to be selected. So I hope that it may work out.
Additionally, I think that students are better off if they have not concluded that academic honors are a joke, and so the high schools should try to avoid letting them become a joke.
I can only tell you about a couple of local experiences: A year ahead of QMP, there was an outstanding student who was not selected for NHS as a junior, because her essay contained a statement of historical fact (truly clear-cut fact) that a faculty member objected to. I was very pleased to see that this student was elected to Phi Beta Kappa at her college as a junior. A few years after QMP graduated, there was a 3-way tie for valedictorian. The valedictorian who was not in NHS was going to Harvard.
You can have a well-run NHS and have either a well administered or poorly administered school. However, I have started to suspect that if the NHS is not well-run, it may be a sign of larger problems at the school.