Looks like useable stuff Michelle. Decomposing at first takes nitrogen, before it gives nitrogen. This looks half decomposed, but it's a mix of many stages. I wouldn't take the chance personally to mix it in all the beds compost. I would try it out in one bed or a part of a bed. Observe the difference.
I use a
straw layer when i start a new bed on grass to kill it off and add a mount of compost on top. This river debris looks like it could perform a similar role and because there are many stick in it they could take longer to decompose and start to act like spunges later on. Which is better than the straw/hay i use. So there is that.
If i have garden waste i just pile it somewhere out of the way, it's a great refuge for some animals that eat snails and the likes, also for snakes, so maybe not good where you are.
If you've got an animal waste stream, like
chickens or cows, i would dump it in their bedding. It contains that much nitrogen, pile it up and compost that, it will make a beautiful compost.
Or use it on pathways between beds, it will compact because you walk on it, keeping it moist-ish and mycelia will enter it, take the nutrients and exchange it with sugar at the
root tips of plants.
Anyhow, very usable! If clean.