I have been waiting on my Family Controls entitlement request to be processed for a little over two months now.
Every week I contact apple support to request an update and every week I am told that there are no SLAs for processing requests for the Family Controls entitlement.
You can see on this thread https://developer.apple.com/forums/thread/725036 that others are in the same boat.
This has been a terrible developer experience - after investing months of work building an app I have now been completely blocked on getting it live for > 2 months. Very, very frustrating.
I will post an update when the status of my application changes.
Good luck to you