I'd scrape the peeling tiles but not worry about the parts that stick... unlike if you were to be remodeling for money.... And then stick something down on top of the old floor. Masonite, 1/4" plywood, vinyl (roll) flooring, etc. would all work. Paint masonite or plywood with thick 2nd coats. I use a half-hearted deep-litter method and a square-shovel to maintain an orderly coop floor.
For scraping the floor: look for a tool that looks like a hoe but not bent. It's sold as a scraper in home improvement tools, not gardening tools. I've worn mine out, I've used it so much. Great for scraping loose flooring. And ice on steps. And poo in the coop.