So I'll weigh in here since I just concluded coaching a 9-10 Cal Ripken team. First place finish, lost in the semi-final game of the playoffs. Yep playoffs for 9-10 year olds. They had playoffs for my 7-8 year old team last Spring as well.
Firstly, I tend to agree with Heine regarding the guys who get involved in coaching kids or on the league boards. The majority make no bones about the fact they are there to take care of their kids and their kids' friends. There are some exceptions but very few. The league president had his youngest kid age out of the league 2-3 years ago but every other board member has kids in the program. They are for the most part nice enough guys, but it's understood why they are there.
I have coached for the past 12 years every age level in the league except T-Ball. I have never had a kid in the program. I'm one of the guys doing it purely for the love and preservation of the game. I have known of only one other guy who coached without a kid and he lasted only one season.
As far as competitive games go, I'm afraid it's what the parents want. I also think baseball is a unique opportunity to teach a kid the dynamics of success and failure and the cause and effect of same while the stakes are still low. There really is no where to hide on a baseball field. If anything the sooner they learn some of these lessons the better. They will be competing all their lives, one way or another.
Now, my compromise is I play the kids at the positions they say they want to play for the first half of the season. I always poll the kids' parents at the beginning of the year and get their first, second and third choice of position. After the halfway point, I counsel them suggesting I do not want to place them in a position on the field where they will not be successful and they should expect to play the positions their skill sets make them best suited for. This is oftentimes as basic as a safety issue. Lack of fielding skill or fear of the baseball can endanger a kid playing a vulnerable position.
Must play rules force me to sit one or two kids for the first two innings. In the first half of the season this was done on a rotating basis. Every kid, including the stars, took their turn sitting.
If all 11 show up for a game, I also have to sit two kids every inning. Same proposition, this rotated for the first half and was more based on merit in the second half.
As far as the pitching question, I give kids a shot if they demonstrate in practice they are able to throw strikes at least half the time. I encourage kids with stronger arms to try pitching. However, I always post a game plan on the fence. It shows where each kid is slated to play each inning. This includes pitchers.
I always counsel the kids penciled in as pitchers the plan could and most likely will, change. This could be due to pitch count limits being reached, effectiveness vs. ineffectiveness, the game score, etc. I have had kids penciled in for their pitching debuts and had to disappoint them. They all eventually got a shot though.
As for putting his kid into an 11-1 game, very bad optics. I would never consider it unless for some reason I was out of other options. However, in my experience a bad pitching outing, e.g. walks everybody to the point he has to be removed, does more damage to a kid than having him wait for another opportunity. I hate, hate, hate having to remove pitchers in the middle of an inning. I want the pitchers to be in a reasonable position to succeed pitching or at any position for that matter. If they don't, it's partly on me.
If your son wants to pitch, practice with him. YouTube has a trove of free pitching mechanics videos. If you're not enough of a player to help him, have him practice with a friend or get him some pitching lessons. If he can consistently throw strikes, he'll pitch. Don't look at it as a right but as an opportunity to be earned.