I'd be more cautious with chickens that were kept in a run with no/few other options but I've always trusted them to choose to eat things that won't kill them (when it comes to natural things like plants, anyway - I wouldn't entirely trust them not to eat metal or plastic!) I know it contains oxalic acid which can be toxic in large quantities and interfere with calcium absorption but so do loads of other plants. I know with horses and cattle there's a risk that they'll accidentally eat things in hay that they'd notice and know to avoid as a live plant. Ime chickens will pick through hay for seeds and maybe eat a small amount of dried grass but they aren't munching on huge mouthfuls like a cow would, so I imagine that reduces the risk too.
My chickens do eat a decent amount of dockens but they'll also ignore it in favour of something else at times, so there's apparently some kind of decision making going on there. They can be very particularly about seeking out specific plants or types of food (seeds, flowers, insects...) at times. They have free choice access to oyster or other shells, so can regulate their own intake of calcium.
Interesting to read in your link that the seeds aren't viable after being eaten by chickens! Dockens can be fairly invasive and hard to get rid of once they're established here, so clearly armies of chickens is the way to go
They're very common here, to the point that pretty much anyone with a little patch of garden that isn't totally paved over or weeded/sprayed constantly will have some growing.
lucky birds I didn’t clobber them! They also ate all the lovely tomatoes that were almost ready to pick. 
