Great read and I can identify. I have a son who was a very good HS player. He started playing when he was 5 like most Little Leaguers. It's absolutely true that you agonize with your kids on the field, especially if they're pitchers like my son was.
One thing I noticed though was that as he got older I didn't get as nervous or as protective. By the time he was a senior in HS and was playing in big time state tournaments and national travel tournaments, I was kind of over it. So I'm guessing that by the time players get to MLB, they're parents are used to it. But I'm also guessing you never really get over that protective parenting instinct.
BTW, my son's 16 y.o. team played against Bryce Harper's team in a wood bat tournament in Fla. when Bryce was 14 or so (he was playing up age levels). That was the first time I heard of him. Even then his legend was beginning to grow.