My son did not pitch in HS (played C, 1B and 3B). He did pitch a little in travel ball, legion and fall ball (local league). I do understand your concern as I knew several of the parents of the kids who did pitch.
Here in VA there are inning limits for HS pitchers. VHSL is still contemplating pitch counts and limitations. Even before the inning limitation (there was a time there was no limit) most coaches in the region did monitor the pitch counts or at the very least the # of innings a pitcher pitched.
Yes, it's sad if you have to monitor your son's pitch count if the coach refuses to do so. Have you discussed your concern with the coach? Have you talked to the AD? I don't usually advocate making an end run around the coach without at least discussing your concern with him first.
FWIW, my son has played college level ball (JuCo and D2) and every coach did track pitch counts whether or not the sanction body (NJCAA and NCAA) required it or not.
As to the score board, do you have a clue how much a decent score board costs? Not exactly inexpensive. Even with a new field (our county opened a new HS to replace the old during my son's sophomore year) it wasn't until his senior year a new score board was installed and operational. I don't recall if the baseball boosters helped fund it or not, but I believe they did help defray some of the costs. I know they did fund most of the improvements to the field (fresh dirt and irrigation system - yes the county did not pay for an irrigation when the field was built).
Your school district is probably like most - they have a limited amount of funds for baseball. The big money maker here is football.
Yes - LL does monitor pitch counts. They don't post it online or a score board though.