My son's experience is that he pitched and hit in HS. He is a little bit different, because he is a LHP, but he bats right. This means that when he hits, his left arm is exposed. First year at JC, they told son he would never play 1B for them, because they didn't want him to have to throw that much when he would be the 1 or the 2 pitcher. They would like him to DH though. He DH'd some, and he had a pretty good year at the plate. Honestly, he would be one of the top 2 or 3 hitters for his JC team, but the PC absolutely does not want him to hit. I believe this stems from a few years back, when a potential 4th round pick was injured on the base paths. (Hurt his non throwing shoulder sliding into second.)
I will add that as others have stated, it is hard to juggle practice for pitching, hitting, and fielding with school. Unfortunately, the pitchers practice separately, so most of the time, you have to stay later or come earlier. With class this is not always possible. I think you have to have commitment form the coaching staff, and if you have a PC that is 100% not on board with his #1 or #2 pitcher hitting, it is probably not going to happen.
It's really too bad, because he is a really good hitter, and they could definitely use him; even if it was only as an occasional DH or PH.
At this point, he has decided he doesn't want to hit anymore, because I think he got tired of swimming upstream. He loves the PC, and I think he doesn't want to upset the apple cart. He also has come to enjoy concentrating on the game that he is pitching, and not having to worry about bad at bats.
When he was being recruited by several D-1's, D-2's, etc., many said they wanted him to hit in the middle of their line-up, and be one of their main pitchers. Looking back, I think they were only telling him what they thought he wanted to hear. I think it is really rare for one of the top of the rotation guys to be one of the main hitters too. When you are 6'4" and a LHP, you are probably going to be a pitcher even if you can hit the ball.