I don't know where your brooder box is. If it is in the house, letting the hens raise the chicks is a huge plus from the dust, smell, and noise side.
If you let a hen raise the chicks with the flock, Mama will take care of integration. If you isolate the broody and her chicks, you lose this advantage.
If you raise them in a brooder, you take care of integration; you feed, water, and clean the brooder yourself. You worry about keeping it at the right temperature. The chicks will probably be more friendly if you handle them daily.
If they are allowed to free range when being raised, broody raised chicks are usually better foragers than brooder raised chicks, reducing your feed bill.
I don't know what size your Cochins are. In case one is full sized instead of bantam, I had one Australorp rise 15 chicks last year. She did fine.
It is a tough call.
By the way, a green egg layer I kept from the hatching eggs I got from you is currently broody on 12 eggs. She is due to hatch July 6.