We have loads of moss in our garden and grass (Oregon). I often wondered the same question. Usually when I'm removing it from my flower beds. I have yet to see any positive benefit to this kind of moss in flower beds.
"Deer and rabbits deposited manure into my gardens and lawn. Still to pull last of veggies and put away to compost. What would happen if I just left it in garden to decompose over winter, pea plants I mean?"