When you have exactly 3 creases meeting at a vertex, there’s no way for them to fold unless the panels themselves bend. In the image you show, notice that the 3 way junctions are all flat, so as a whole it’s effectively acting like a panelling of equilateral triangles, but with decorative lines to the centre of each triangle.
Not that this is a problem, but something to consider when designing patterns - the parts connected around 3 way vertices will act like one rigid panel.
If you’re looking to create the sort of crumpled effect of these wood textiles, you don’t need to decide which edges are mountains/valleys - you can just fix all the edge lengths and see how it deforms as you pull it with grab, or drape it over something.
If you want to include the quads as flat panels, you’d need to include something to keep them planar (either Planarize or just Length goals for both their diagonals).
For a more flexible result though, you could triangulate everything, then all you need is EdgeLengths.
I don’t know how this would behave on a larger patch. There might still be regions that would lock up and move together, but the easiest way to find out is probably just to try it and see what happens. There might be some interesting effects from the lack of continuous straight lines or translational symmetry.