I don't see "family time" on that list.
I don't know what to do.
Make hard choices.
Every one of these groups is run by people who think that the world should revolve around what they do. Were I you, I would set aside a particular time for family, daily if possible, that must remain inviolate. Else, it won't happen. Then I would enlist the kids in deciding what has to go or what activities they would do together; IOW swap the gymnastics for dancing or whatever. Remember, if you kill youself or induce serious family strife by trying to do too much, you took something more precious from them in the process: yourself; i.e., your time with them. Any of those activities, by comparison, means far less.
I would also suggest that you lead some of those activities instead of relying upon an organization. Take them to museums, or even take a course with them. Show them what "mom likes to learn too" looks like. They'll learn more from that than you'll realize.
My husband's a Cub Scout leader. I'm a Brownie leader. I lead a Bible discussion at church during the classes 3 of the children attend. (I read all the time, but I can't take classes with a nursing baby!)
You're right, there are hard choices to be made, and if I'm ever at home at the same time as my husband, we'll have to discuss it :-).