Well I only read a few pages of the post.
I am speaking from another sport point of view I guess (figure skating).
I watch all my daughters practices and competitions. She is still very young and its not comfortable for me or her to be left there by herself (she is almost 5). But I do like to see what she is doing and what I am spending my money on I want to make sure its not wasted
As well school may be more important but there is not as much risk of injury. I am sure when my daughter is 8 she wont care as much if I am there when she gets hurt but for now I wont miss a practice.
Also I am unfortunetly one of those parents who has to drag along younger siblings and make them sit there while she does her thing. If i paid for a sitter every lesson or event my daughter wouldnt be able to participate in the sport. I make sure they have things to do or eat while sitting in the stroller and when they get older they will have their own involvements.
And on the note of parents "coaching" during open gym. It depends on the level of coaching. I find open gym can also be a chance to practice if wanted not just play time. And if you are paying for your child to practice there is nothing wrong with making sure they get in some practice. My daughter gets to go to our open skating sessions. I tell her what she needs to practice (coach's list) and then once she does it she is allowed to goof off and play. But if I am paying for it she needs to at least get something else out of it. I could go to the park and play. But I dont find this as being too involved.