Well, if it's free of weed seeds, then at the worst, it'll just be a source of organic matter for your lawn. Not true compost though, just organic matter. True compost is black, and doesn't resemble tree bark, to me.

If it's a new product or enterprise, I'd probably pass on it and see if they survive.

All the tree bark and wood chips I've seen have been brown, but I can't say if green is impossible.

There is also a compost forum here, where they might be able to say more.

mnten, if you can identify the original material, leaves, twigs, etc., then the material is not completely composted.
An essential condition of the process is that the pile should "cook" (140 deg. F). The pile should be turned inside out to ensure that the outside layer gets "cooked".
This cooking destroys harmful organisms and seeds.
Finally, composting is neither quick nor simple but immensely interesting, satisfying and beneficial.

It sounds like the equivalent of shreded leaves, in other words, added organic matter, not "true" compost.

My inclination is to add it. Most urban soils are low in organic matter. So long as you don't smoother your grass, it should only help. Worst case, which is not likely in my opinion, you might tie up some nitrogen in the soil as it in essence breaks down in the soil, requiring a bit more nitrogen fertilizer than normal. But, I really doubt it.

It also sounds like an ecological solution to not adding these end products to a landfill. Good for soil, good for the environment. So, imho, good stuff, go for it.

I agree with Dave here. You are bringing in weeds even if you do not see them. Also bugs (good and bad)...careful