Full disclosure, I'm in beautiful sunny Florida (where we average 54" of rainfall per year).Thank you Thank you! My instincts agree with you. I just wanted to go on more than that.
The only parts of my builds that aren't PT are made from Hardie Board (a fiber cement product) or steel roof panels.
Modern pressure treated stuff isn't what it used to be, just like the paint on you car doesn't hold up nearly as well as the old lead-based paints they used to use in the 60s and early 70s. Some of the new PT stuff isn't even rated for ground contact!
But your chickens won't spend any time eating it, even if it were full of copper sulfate and painted in creosote. Nor do you need to worry about off gassing. Unless your coop isn't properly ventilated. Which is a MUCH bigger problem than your construction material or the paint you put on the outside.