DD can do a lot of that kind of thing (although she is only 2, so still getting there with some things), BUT I have to fight my natural instincts all the time as it doesn't come naturally to me to let her do risky stuff.
An example is that there is a high horizontal ladder at the park, and she walks across it. It's higher than my head, and she's confident in going over it. She's slipped a few times, but always caught herself by holding on to the rope handles that go across. Her legs can't reach the rungs as the stride is too long for her, but she walks on the sides of the ladder. I stand underneath in a "ready to catch you, but acting very cool with it like I'm not there to catch you" way.
I think it's important to let her do those things and build her confidence, but I find it hard to do.
Each to their own, but for those who admit to being a bit more protective, I would very politely say that they may want to think about not saying to their kids (when they're older) anything like "Well it was because you were so klutzy" - or anything along those lines.
I'm sure you probably wouldn't, but I say that because I was very overprotected as a child, and my mum always said (and still does) "Well, that's because you had no common sense/ would have been the one to hurt yourself/ would have got lost/ would have fallen off, etc."
When I met DH I had no confidence in my ability to do any kind of physical task, and would always put myself down with things like that. It took DH to point out that I actually wasn't completely hopeless at everything, and that it was probably a result of my parents saying that all the time (that I thought I was).
Anyway, a side note, and not trying to tell anyone what to do. I just saw a few comments like 'Well MY DS/ DD would definitely hurt themselves" and it kind of brought it all back to me!