Development of any kind is a shared responsibility. Generally, there are 3 or 4 key people when it regards the development of any human being in any endeavor. It simply drives me a little batty whenever I see people raising up showcase baseball as the end all be all, when it's simply part of the puzzle. The programs I am familiar with, and some of them are elite...play games on the weekends, in the summer and fall. Period. This is not a bad thing. They are very good at what they're asked to do, and that is to showcase their players. They also have very talented and qualified folks leading them, and certainly can play a huge role in the outcome for the player.
Regarding who spends the most time with a player...if you are part of a solid HS program, you may spend the better part of 9 months per year from as early as the 7th grade with that staff. (As much as 56 months). I consider that to be fairly significant, and frankly, other than Dad, I'm not sure who has spent more time with the kid.
You live in a different world than I do. We are faced with very stringent restrictions on when teams can practice. They basically start the 1st of March and run till end of May (so three months). Then they are allowed an "optional" summer season, which is basically June. Most of that time is devoted to games (not a lot of development for those on the bench). Other than that, it is hands off. They run "open gyms" in the fall and winter, but the coaches can't instruct, just observe so nobody gets hurt. And if you are doing another sport, you can't participate. So there is a relatively short time that the HS coaches have access to the kids. Add the fact that with 20 or so kids on a team and two coaches, there is very little opportunity for individual instruction.
So most skill development occurs thru the travel/showcase programs. Many have a focus on off season training - hitting, fielding, pitching, catching, etc. Many players get hooked up with skill specific coaches who are part of the program. It is the thousands of reps with these coaches where the kids develop.
So unless your high school coach can have access to the kids out of season, it is difficult to develop players.