As a matter of practice I am a huge proponent of demonetizing youth sports at every level. In the area of private instruction I am willing to risk the humiliation of hypocrisy. If you want to excel at the highest level possible or even compete respectably in travel/club ball, then private instruction is a necessity.
You can effectively train your own child in the early developmental phases (8-10 years old) however beyond that window you rapidly become nothing but white noise. Naturally I concede there are exceptions but they are far and few between. Even if you possess the necessary expertise most children become very comfortable challenging the parent in this setting, where they do not feel comfortable challenging another instructor, thus making the training more efficient and effective.
Throwing and catching in the yard absolutely helps with repetition, work ethic and muscle memory, however it lacks the application of specified techniques and mechanics required to elevate the players level of competition. Hence we go into our pocket and pay someone to get the job done. Parents push back on this all the time, but they are just avoiding the reality of current times. Yes your Little League and High School coach taught all you needed to know, but back here in reality kids are being taught things on a daily basis that were cutting edge and only being employed by a few back in your day. By the way this is not just a softball thing. It is a whatever interest your child has thing.
The refined art of pitching and hitting is so integral to the fabric of the game that one would not survive past age 12 without professional intervention. This same intervention of survival is and must be maintained throughout the duration of the players entire career.
Sadly this reality has had some negative repercussions on our younger generations. Unfortunately younger players are confusing the importance of personal training time with the time they spend in private instruction. Simply put they feel the weekly lesson is their training. These players are failing to understand the lesson is meant to instruct and teach, where the daily workout is meant to reinforce what is being taught. Success is realized from the time spent building on the lesson and not from the lesson itself. If this perception is not changed quickly, the quality of play at the highest levels will deteriorate rapidly. Many believe this has already occurred in the circle. Arguments have been made that the quality of pitching today does not measure up to the pitching of the 70's and 80's. The reasoning, less and less time is spent refining this very difficult skill set.
I am often asked who should I bring my daughter too for hitting, fielding or pitching. There is never a loss of people willing to offer up the BEST instructors in your area. I consistently offer the same advice, particularly for younger first time students. You can go with the instructor who has been dubbed the "BEST" and you may have great success. However you may go with the "BEST" but if that instructor and your player do not gel, then it doesn't matter how good they are. If your player is miserable and hates going to the lesson, then you are just throwing good money out the window. Take the time to find someone highly recommended but do not commit to a schedule until you can gauge how effective the relationship between player and instructor is going to be. Additionally do not get caught up in a long term relationship with any particular instructor. Even the best relationship has a shelf life. There will inevitably come a time when even the best instructors have given all they have to a particular player. Be vigilant and ensure your player continues to make progress. When progress has stagnated, it may be time for a change.