AITAH for telling my employee she can't bring her child to work anymore?

So, I work at a summer camp as a camp director but during the schoolyear I work as a teacher, so I often ask my co-workers to work during the summer as many teachers like the extra paycheck. We work at an elementary school in an urban-area but the summer camp is in a more affluent area of our city. All of this context is needed in this story. So, I hire one of my co-workers, we will call her Sarah (Grown Woman, Married) and I let her know we also allow our employees are allowed to bring their children to the camp to make it easier for them. June was fine, but July has been a little rough. We recently had multiple parents complain about Sarah yelling at the children at the top of her lungs in the gym. I had not seen these incidents due to being on vacation for my birthday recently, but it was brought to my attention by my bosses, the owners of the gymnastics gym that the camp is ran out of. So, this lead to me having a very uncomfortable conversation with her today because the problem is she was yelling at her own child.

Her son, we will call him Duke (3M) , has been terrorizing our camp, running out the door away from his mom, screaming and crying constantly, not listening to his mom which has led to Sarah having to get stern with her son. But, when you are yelling at a child in front of parents who don't know that is your child, it look very concerning.

This lead to me having to call her to find out a solution. I tell her the complaints we are receiving and what my boss has said. I don't want to let her go, so I suggest that maybe Duke can't attend camp anymore because it is taking away from her ability to focus on her job by having to yell at him all the time. This is best solution I could find because I was hoping that if her son wasn't attending the camp, she would be able to relax and focus on working and not wrangling her son. She starts to cry and I tell her that if she can't find someone to watch her child then we can find a different solution but to try this for now.

She now has texted me saying "she can't work somewhere where her child isn't welcome" and that she is quitting for the remainder of the summer. Which is odd to me because I never said that but when you have to yell at the top of your lungs to get your child to listen to you, we can't have that at the camp. Now, I have other employees saying that she's pissed and that I'm the asshole. So AITAH ?