I was head hunted for a full professor position here in Australia. I talked with the headhunter on the phone and also in a face to face meeting. These were like regular phone or conference interviews. I didn't make the cut for the on campus interviews. The guy was an education specialist but mainly dealt with administrative hires I think. He didn't have too much clue about US academia and some other subtleties. It was an interesting experience.
That sounds very similar to this setup - initial round of interviews, headhunter does a lot of education work but includes admin/VCs/school head teachers.
Even just having got to this point has likewise been interesting, to be honest.