I would be concerned the chickens would eat the paper and it would create an impacted or sour crop. Not a problem with this? Then again, paper is derived from wood, which they eat all the time.
I have never had a problem with chickens eating paper shreds. I use paper shreds for coop deep bedding and shredded cardboard for the nest boxes. It's everywhere, and as such, I think they just treat it as litter. If they did eat a bit of paper shreds, I doubt it would harm them.
I just don't worry about it. Like you said, they eat wood. I'll add they even eat rocks. So, I just don't get too excited about a chicken eating something like a few paper shreds.

I have heard some people mention that they tried using paper shreds and they thought the chickens were eating them. So, they were uncomfortable using paper shreds. I imagine that if the chickens have never seen paper shreds, and only a handful of paper shreds get tossed into the coop or chicken run, they might try to eat a few to see if they were any good as a novelty item. I have yet to hear that any chicken will eat so many paper shreds that they get impacted or sour crop.
Having said that, I have chicken feed available 24/7 in a 5-gallon PVC hanging bucket feeder inside the coop. My chickens all seem to know the difference between chicken feed and paper shreds.
I also brood my chicks on paper shreds litter. It's a great, cheap brooding deep bedding litter. I fluff up the paper shreds and/or toss some fresh shreds into the brooder as needed for the 8 weeks they stay in there. Then, the paper shreds get tossed into the chicken run for composting when I move the chicks out. I have never lost a chick due to eating paper shreds in the brooding setup, and that is where you would think would be the highest chance of any potential negative effects. But, even as chicks, they all seem to know their starter feed is good to eat and the paper shreds are just not interesting to them.
Although paper shreds have become my favorite bedding material, I am not opposed to other materials as well. I have also used free wood chips, dried grass, and leaves, for example. All free resources for me. Many years ago, I bought straw for litter, but my free options just work better for me. I also have used sand, because I live on a lake, but that always seemed to get dirty and smelly really fast. I had to clean and replace the sand about once a week. With the paper shreds deep bedding I now use, tossing on fresh paper shreds about twice a month, I only clean out the coop 2X per year.