I have a friend who design playgrounds, and he operates on the principle that there's two kinds of safety. It's the objective safety I mention above, and there's the subjective safety which you mention. Kids should have objectively safe playgrounds, but they should be allowed to challenge their subjective safety feeling.

The objective safety has mostly to do with the design and maintenance of the equipment. The subjective safety is more due to the design of the playground itself and the elements in it.

One example is a climbing wall. With the right layout, sufficient impact sand or rubber on the ground around it as well as proper maintenance, it can be objectively very safe but still allow kids to feel they push their boundaries.