viking - I think you raise something interesting.
I coached a team in Cooperstown. Similar scenario - kids had been together (more or less) for 3 years. Parents and kids mostly from an affluent area.
We decided we wanted to play everyone about the same while still trying to win some games at least...and we did fairly well. But some parents did begin counting innings. And while there wasn't an issue of my kid/coaches' kids playing more, they did count things like innings at DH vs. in the field.
A couple of them approached me, they were reasonable in their approach. I listened and tried to adjust as best as we could.
The difficulty was that more kids were good outfielders than infielders. So sticking a few kids in the infield was gonna cause problems and not just winning problems, kind of embarrassing problems for the kids themselves in fielding ground balls well for one. The DH was, in effect, a way to get a 4th outfielder in the game at any time and I might be mis-remebering but I think at that time in Cooperstown you couldn't rotate your your DH into the field and swap an OFer to the DH which many tournaments allowed.
Anyways - I don't think what you asked, as you described it, was out of line. You spent your money and your kid, unless on a team designed to win at all costs, deserved to play his fair share IMO.