If you get rid of the brown particles in the chicken bone broth, are you getting rid of a nutritious part? If so, how much of the broth's nutrition is in them?
The brown bits in a broth are the pieces of meat and skin that underwent some nice changes, the maillard reaction, during roasting or pieces from the marrow of the bone.
The broth has been simmered for some time, so most of the nutritious and delicious compounds are in the liquid. Most, that is, but not all. Some small amount remains in the flecks.
You can strain it for improved color and visual appeal, or leave them in. You can certainly filter some and do a taste comparison, but that does not speak to 'nutritious.'
Basically, all the work to roast and simmer means the broth has the good stuff. The flecks are a little bonus.