Uncoated plywood doesn't breathe either, but it might absorb some moisture.
Always good to measure humidity inside and outside of coop, they should be about the same. If more humid inside, ~10%, you need more ventilation.
Hard to assess draft in that coop as you can't get inside and no windows to observe wind ribbons.
If you mean OSB plywood panels you are correct but regular plywood does.
I pulled this from web. Off this site
https://www.ecohome.net/guides/2190/which-is-better-osb-or-plywood-sheathing/
“
Plywood will absorb moisture quicker than OSB, but consequently it will also dry quicker. Both products will swell when wet, but plywood will return to almost its exact shape when it dries. OSB is less forgiving of being left out in the rain and will show more deformities after a wetting and drying cycle.
Beyond concerns about exposure to moisture during construction, is the moisture that can infiltrate through wall assemblies after completion. How durable a wall is depends heavily on its ability to dry, and how well it can dry depends on the materials you chose for your wall assembly, your 'recipe' if you will. ”
Edited to add this bit about permeability.
“The vapour-permeable elephant in the room...
The most commonly held opinion among building industry experts that we have read and spoken with, is that OSB has a perm rating of 2 and plywood has a perm rating of 10.”
So a vapor retarder yes.
This has been my experience as a home builder also.
Often time venting of a home is done along the lower edge of roof under the overhang of the roof and with the same amount both sides with a vent along the ridge.
This gives the best air flow across the under side of roof.
Good for moisture and excessive heat build up cause by sun baking the roof. Which may not matter to the chickens but does make the roof last longer.