The short answer is no, you probably don't need one for every skill. Here's how to explain that to a parent who asks.

Obviously it depends on the seriousness, age, talent level and other factors whether a family should hire even one coach, or several different coaches. Often, parents who are very busy and have the resources to provide specialized training in many areas fall into a normal flow of hiring a private coach for every single activity their child does. This is not good.
Seven, eight, nine and ten-year-olds are not wired to have individual relationships with five, six, seven or eight different adults. So what's the answer? It depends. If a parent has a child who's showing an interest in one sport, say baseball, then it's a good idea to find a young, fresh, passionate coach who has committed to that sport for their whole life, whether they played professionally or not.
01
At CageList we focus on growing quality coaches, with communication as the primary factor in deciding who we allow to work with a player. The majority of the players we work with are under 15, and the vast majority are under 10. This is where most private coaching lives, in the youth world.
Parents think it's beneficial to hire a private coach or specialist for every different activity a kid does, at least most of the time in our experience. The majority of players we know who get private training for multiple sports or activities are generally very soft. They don't have toughness, they don't listen well, they aren't coachable, and they're always looking for answers from other people rather than ever having to figure it out themselves.
If a kid isn't tough (if they don't have thick skin, or can't manage themselves at least to some degree), that becomes a primary factor in how well they learn in sports and in life. If a family doesn't remember that toughness matters, they're always going to be more detrimental than helpful for the player in the long run.
02
If a family has already hired five, six or seven coaches for different sports, they've made that choice and it's not your job to criticize it. But the more valuable path, for the player and for the family's budget, is usually one quality individual or a small team of quality mentors, working with the player as a unified group rather than seven separate specialists.
A coach who's played at a high level in one sport doesn't need to have played football in the NBA to teach a kid to play defense or take a proper stance.
03The argument
It's making sure the player's mental development as an athlete is being handled properly. That's the argument worth making, and it's a true one.

Isaac Hess founded CageList after seven years of professional baseball and a coaching career of his own. These are the principles we share with every coach on the platform. They come from his academy, MADE Baseball, CageList's sister company. More of the story at isaachess.com.
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This ties directly into how we think about the value of a single lesson.