I agree that as a society we are not as demanding of our kids as when we were growing up, but I wouldn't say that is the case with parents of athletes. If anything, we can be a bit too demanding at times.
With that said, no one is saying that we have a problem with babying our kids. The consensus today appears to be that we have more knowledge about overuse injuries and there are things that can be done and things to look out for that can minimize these long term effects. Softball pitching is not as damaging to the arm as baseball pitching, but there is damage to the body that can accumulate with time.
Our daughter now warms up slowly and gradually and makes sure her warmups are long and loose. She has also worked very hard to make sure she has good mechanics and understands what she is doing to keep her mechanics sound. She works at keeping her spins more in her wrist, and less in her elbow and shoulder. She also believes in icing after long pitching outings to keep the muscle inflammation down (both elbow and shoulder). In fact, her college trainer has all pitchers (and catchers) do full body ice baths after a long day (sounds painful).There were two times in her travel ball experiences where she couldn't pitch. One was after a long day of multiple game pitching (Best of the Best) that ended in an extra inning thriller late at night where she didn't come out of the game and gave it her all for the win. (She also didn't ice that night.) The next day she couldn't throw. Probably wasn't the best for her long term health, but she did recover. The second time she didn't drink fluids during the day as her mom instructed, and she paid the price (heat exhaustion). She now listens to her body much better (and her mom).
I think that every pitcher is different, but all can follow some general guidelines that can help maintain their health (such as icing). One other key is to be observant. Look for good / bad mechanics, soreness, fatigue, etc. Know the athlete and how they respond to high pitch counts. And gradually increase the amount of pitching they are capable of as they get older. You can use pitch counts to track general trends, but I wouldn't recommend setting arbitrary limits on pitch counts like baseball does. I think CoachDennis's reply summarizes things well.