Many of my quail are broody, but I always take their eggs, I don’t think a community pen is a good place to raise chicks, plus if they flush and come down on the chicks it’ll hurt or kill them. My indoor quail readily accept chicks and care for them. New babies are too small though, I had a situation where the adults got startled and flushed and one came down on a chick, and he was smashed into the bedding and stunned, I thought he was dead but he came around. I took them out until they were 5 or 6 days old and they can easily avoid danger now. This is their 3rd set of chicks I think. It makes both the chicks and adults so happy, and my male is obsessed with them.
Quail have so many young because the mortality level is so high, even in a pen without predators, they can easily get lost from their caregivers or squashed. But if you want to encourage broody behavior, you should keep them on the ground, with deep and/or natural substrate.