I agree with some that say the kids are learning a lot of what chores may teach through their gymnastics (responsibility, etc.) but one key component they don't get at gym is the family connection. Yes, gym can feel like family but it's not. And in our house committments are 1. family, 2. school, 3. gym (or other activities) so it's important that our DD participate in household events and family events as well as do her part around the house to keep our family home in shape.
She is an only child so we don't deal with the struggle of what her part is compared to siblings, but I think we are on the moderate side of chore lists (definitely not light but I don't think it's too heavy either) that include both daily things like setting and/or clearing the table, cleaning up after dinner, picking up after herself, etc and weekly things like cleaning her/the guest bathroom, dusting, vacuuming and helping with yard work.
If she can't find time in the week to complete her chores then she skips her optional gym time to complete them, no arguments. There are occasions where we've let her out of a chore for a week or whatever because of too many conflicting things. Like I did some of her chores the first week of school because getting into the new schedule was a challenge.