Not defending High School ball or any particular coach, but is it possible some of the coaches are doing what they can with what they have? Even if you have 3 or 4 possession players (which most non-private schools would be extremely lucky to have), you can't really play a possession game (unless you plan those 4 players taking the ball up and down the field - and even then - someone else will eventually need to "kick" it to them).
So, you end up with what you see. That doesn't mean the coach is bad (it doesn't mean they are good either). At this point, it's clear that High School ball is what it is - a social event. Which is important, as someone has pointed out.
But, if by 16 years old, your child can't adjust between 2 teams style of play, then either they aren't good enough, or shouldn't be doing both, so as not to confuse him. I would always tell my child to "play their game," but listen to the coach. If my son has superior foot skills, he should use them within the framework of what the coach is telling the team to do. Just because it's kick and run, doesn't mean the coach just wants them kicking as soon as they get it. He can still posses and look for an opening/teammate to place the ball. Direct is a style of play that is well accepted internationally.
Also, don't presume that a bunch of 16-18 year olds are actually listening and executing everything a coach is trying to tell them. Yes, it's part of the job to figure out a way to get the kids to do what you tell them. But, as a parent, does your 17 year old also listen and do what you say to a T? And does your 2nd 17 year old react the same to the same methods? Now, extrapolate that to 15 new kids each year.