In Dad's defense, he had to use treated wood to avoid any possible rot at the concrete/wood interface. I've also demolished a deck that was infested with termites. You wouldn't want that either.
But as far as a danger to the chickens, I think that is very remote. The use of arsenic in wood treatments was
banned by EPA in 2003, and it was the arsenic that was most problematic for leaching out where you didn't want it. The new treatments use compounds containing copper, boron and various organics that, when they do leach, are much less problematic.
Plus chickens usually don't scratch on wood looking for food, that would be woodpeckers. Chickens are dirt-scratchers, and any wood they would ingest would be in the form of sawdust. If you do put the chicken run under the deck, I would suggest putting down a 4" layer of wood chips. As the fungi decompose the wood chips, they will be breaking down any wood treating compounds that leach out and drip-drip down. Plus all those wood chips will harbor a lot of insect life that the chickens can chow down on. There have been many threads here on Permies about how piles of rotting wood chips make great buffets for chickens to scratch through.