Not a fiber playground, but in Belfast Maine there is a playground that is pretty natural. They have a block fort like what was pictured in one of your links, but smaller, with huge boulders arranged for kids to play on, then a tunnel to crawl through, a slide and swings. They have a hand pump so kids and let boats go down a
water trough, with little A framed cabins to play on. Between where you park and the playground, there is a bridge over a stream that draws kids like mad. If a family wants more, they can wonder down the back way to the ocean part and play along the rocks and gravel that makes up the beach as the tide goes in and out.
Its not 100% natural, as there is some steel to make up parts, but no plastic to speak of, and pretty darn natural for a public playground.
But its a pretty darn big park, so the rest of it has tennis courts, ball courts, work out equipment, dining pavilions and that sort of thing, even a swimming pool. Its actually pretty good, and all free. It is definitely an asset to the community for sure.