I did a bit of playing with this. Swapping a few traces between top and bottom layers, and minor route changes got rid of several more vias. Without repositioning any of the parts.
Moving parts around a bit could clean the routing up more, but I know nothing of the physical constraints you are working with for the headers and LEDs.
I ended up deleting a bunch of redundant bend points as well.
Untitled Sketch 2.fzz New-thin-traces.fzz (57.1 KB)