I think it is the father's decision how he and his family will organise their time, while he has contact. I don't think it is the child's decision, or the mother's; their preferences do not automatically take priority over his.
The child's preferences are secondary to the father's! The father's take priority. Seriously? What school of parenting is that from? The child is not a possession of the father or the mother, he is a whole distinct human being with a whole separate life, even at the age of 6. Yes the father can decide his son doesn't eat meat when he is with him, or always has sunday dinner or whatever but for a father to absolutely ignore the important content of his child's life - school, swimming etc. is awful. And yes he can do it . But it doesn't make it right and it doesn't mean the mother in this situation shouldn't give her best shot at doing her best for her child.
Do you really think a father's preferences should be unquestioningly given priority over his son's actual day to day life - the content of his life, the things that form his childhood, just as they do everyone else's-friends, school, competitions. This is a chilling view of child rearing post divorce.
Yes, it does suck when courts get involved with issues after a divorce/split. It is the price you pay unfortunately. To get to the stage where a judge is deciding whether your kid can go to ballet lessons, that kid has already probably paid a much worse price than you.
And many many parents manage to organise themselves even after a split so they don't end up fighting over everything. I suspect they are parents who worry less about rights and priorities and put their children first.