That's what people do, or should be doing, along with the whiteboarding. I don't think anyone just has them do fizzbuzz and that's it, like you seem to be implying. That won't even fill an entire interview slot.
> it seems like it is much more enlightening to hear about how they organize a schedule...
Sure, and that's also something you ask about, but that doesn't weed out the people who can talk the talk but can't really program, who do exist.
> But even as I type this, I'm wondering if the ability to read people is why this so hard for software interviewers...or maybe it's for lack of that simple skill that our industry has resorted to silly proxies like fizz buzz, etc.
I think it's because people want real evidence of skill, which is pretty easy to detect via whiteboard in most cases, not just the interviewer's gut feeling.