Teachers also have fairly strict licensing and educational requirements (and get paid squat).
I know a lot of finance people. These are not the most stable or reliable people on Earth, but they have to pass their Series 7s, in order to work. They tend to be kind of a scruffy lot, otherwise, but they make a lot of money.
To the degree that it's a reasonable thing to ask in terms of time, it's hard to argue against asking someone to demonstrate something that they'll actually be doing on the job. Certainly I don't think, in all but the most extreme cases, that anyone would argue that there don't need to be auditions for actors or musicians.
When I was an IT industry analyst, we did ask for writing samples. At least one of the big firms did require at least some candidates to prepare a presentation on a topic and deliver it. (This is an example of a sub-industry that has very few formal requirements and people come from all manner of different backgrounds.)
My fiancee is a high school guidance counselor, and she essentially did a demo session with some current students during her interview.
I don't like the situation we are in, but I think your examples actually help illustrate why things are the way they are: we don't have a gating mechanism into our field. Most of your other examples do.