You are only gonna girdle the tree with direct friction of rope on bark, like if you wrap the rap rope directly around the tree.
A sling tied around a tree will not "girdle" the tree, because the sling will rot or snaffles will eat it or someone else will replace it. I have encountered plenty of manky old slings around trees (some 20-30 years old on obscure routes) and have never seen a sling that the tree has grown into and been girdled by... the only way it could happen would be with thick hawser or cable or chain, I think. ( I have seen this with old chains people were using to anchor a hauled-up-on-the-beach boat to)
So then, the setup Daler describes does not benefit the tree the way he purports it to....
Why NOT use it? Well the only real disadvantages I can see are that it uses 2 knots instead of one, so it requires a greater length of tat than just a simple loop around the tree... also two knots is twice as many potential points of failure as one knot, so it's a tiny little bit less safe.
Basically I won't be using it cause its a case of aint broke dont need fixing.