It isn't about AA or her recovery. She is a stranger basically. If she has a childcare business ask for references. Just because she is an elderly grandma doesn't mean she can take care of kids.
As a mom with a special needs child, I totally get it. Has nothing at all to do with mom being a control freak. My mom is elderly, not in AA or an alcoholic and I wouldn't leave my son with her because he is just too much for her to handle safely at times.
Has zero to do with her recovery. It is called good parenting skills you have.
He is wanting to prove something to you about his recovery by using the childcare issue instead. Don't question yourself on this issue.