Given their respective schedules, there is no one time that all 4 major sports are off. However, right now, the NFL is off, baseball really hasn't started and the NHL is beginning to wind down (do the NBA and NHL have incredibly long post season play or what?). The offseaon would be a good time for divorces, custody matters, whatever issue to move forward. You don't have to worry about games and practices.
Non sports folks don't get an offseason when things can be scheduled. Nor do they get to ask the court for special consideration. The court sets the hearing dates. The parties just have to tell their employer they need the time off for court. Then there is meeting with the attorney for discovery, trial strategy, etc. Depositions need to be scheduled. Attorneys like their weekends too, so we tend not to schedule around our client's schedules. All in all, a court matter means a lot of time missed from work.
What is the solution? Don't get a divorce? Don't fight for custody of your kids? Hardly. If you must take your family law matter to court, do it. Accept the timing and do your best. It is all one can ever do.