It depends not just on the club and the insurance but also on the activity.
One club where I work allows kids age 8 to take cardio classes with a parent in attendance. They can attend independently and have freedom throughout the club when they’re 14. But if they want to do a weight training class, they must be 12. The other gym where I work, it’s 12 in group-ex and 14 for the weight room floor. The place where I pay for membership, it’s full reign of the club at 13.