My kid is that age and he has been taking lessons since he was 6. The more lessons he takes from somebody that knows what they are doing, the better his swing will be in my opinion. He doesn't take them on a regular basis, just when I think he needs them. The older he has gotten, the less lessons he has needed. I've also learned a lot and can now see issues and try and make him aware so he can correct them.
At 11u now he does tend to drop his hands. He is aware of it and at our last hitting session with guys at our academy, he came over and asked me if he was dropping his hands. Good habits will pay off in the long run. It doesn't hurt that he is able to dissect and fix his own swing. I would hate to have him continue to do something wrong for 2 or 3 years and then have a hard time "breaking" the bad habit. At this age, a lot of times coaches are not really experts. They may have played high school baseball, but that is not the same as someone that has played MLB baseball and knows something about hitting at that level.
I have a 6 year old daughter, she has asked to play baseball like her Bubba. Okay, no problem. But the thought of paying for private lessons to tweak the squish the bug concept from a MLB player for a kid who still wears pull ups at night because she can't wake up....what is the dictionaries definition of wasteful?
By the way, my son also knew what his body was doing wrong in a swing by age 11, but that was through his coaches telling him, not paying for extra lessons. If the coaches don't know how to swing a bat you have the wrong coaches.
I understand what you are saying, just do what you feel is best. At 6 no kid will have a big league swing. At 10 there is a difference between a kid that has been taught properly vs. one that has not. All of the coaches for my son's team are dads. We don't have paid coaches at this point. Most played high school baseball, I think one of the assistance played in college. The coaches basically run practice and put together the schedule. I think they have learned a lot too over the years. They learn from talking with other coaches and picking up what the "experts" at the academy do with our team during workouts. The more work we do with the academy guys, the more expensive our fees as it is built in. I do however supplement by getting my kid pitching lessons at the academy as well as taking him to see a former MLB player for hitting lessons.
This is my philosophy, if I wanted my kid to learn karate, I would get him karate lessons. If I wanted him to learn to play the piano, I would get him piano lessons. I think most anybody can run, but if he wanted to be a sprinter, I would get him a track coach/trainer. Baseball is no different. When he was 5,6,7, sure I was out there with him in the backyard practicing how to hit off a tee, off of a pitch, learning to throw, learning how to catch (we still do this). He progresses beyond my basic teachings quickly. Just like with pitching, mechanics are important, that is why guys have coaches. I don't think its ever too early to get a coach once you have the basics.
From what I can tell, a proper swing is something that has to be built. Young kids naturally "cast" the barrel when they swing a bat the first time. The drills that are done to help you understand what keeping your hands inside the ball are important. I know in my son's case they have paid off to date. If he wants to try and play baseball as long as he can, he only gets his youth once to play and get better. I would like to do all I can to help him be the best he can be. That includes putting the right folks around him.