Higher ed, in contrast, is less credentialing driven, so while a master's may help, it is not a slam dunk.
That's a good point. I don't know anything about K-12 administration jobs, so what I said up-thread only relates to higher education student services jobs.
A lot of higher education administrations jobs require a certain level of education and a background that is in some way related. That's why job ads will often say, "a master's degree in counseling, student affairs, educational leadership, social work, or related field..." My department does require M-somethings, but we've got an MSW, a MLS, several MEds, and people with an MA or MS in an academic field.