Gaps aren't as bad as I feared but I would still cover them up somehow. I lost a chick that got her toe caught in a gap maybe slightly smaller than that, and she died from injuring herself in a panic.
I agree that toes catching would be a concern. They could catch in the existing gaps too so that's something you want to eliminate as much as possible.
I would do the plywood. Or scrap linoleum, if you want something more waterproof.
Not sure why drainage is a concern (you said you want...