Since I've been on the Board of our local kids rec league for 17 years, I have a lot of experience with both the timing of games and the playing time for the kids. We restrict the number of players on each team to a maximum of 10 and any kid who shows up for both practices during the week must play at least half the game on Saturday. That means that if a team has 10 kids and they all show up for the practices, they will all play half the game on Saturday (or on a weeknight for HS level). Of course, this timing is somewhat approximate, although as close as the clock situation will allow. Our 3rd and 4th grade teams play 8 minute quarters with running clock, 5th and 6th play six minute stop time quarters, 7th and 8th play seven minute stop time quarters and HS plays 20 minute running halves. This is dictated by the amount of time we have to get all the games in. HS games are played on weeknights. We rent the gyms from the School District and they are pretty strict about us not running over the rental times because there are custodians in the schools who stay in the gyms during the games (school district policy to have a school district employee in the gym during games) and they are union employees so the district is very specific about their clock in and out times.
BTW - the referees are not responsible for monitoring the amount of playing time the kids get. Our policy puts the burden on a parent to complain to the Board and then a Board member checks it out. Since I am a referee
and a Board member (as are a few others), I sometime get asked to "testify" as to my perception of the playing time of a player if there is a complaint but usually we are not concentrating on keeping track of this. We have enough else to do smacking unruly parents upside the head with frying pans. You should try it sometime. It works pretty well.