Perhaps I am the only one, but I have always seen Charles as being EXTREMELY spoiled and 'do as I say, not as I do'. I think the reason for his push for the slimmed down monarchy is 1. because he knows that the Commonwealth will implode the second QEII dies and 2. that he is very territorial of 'his' rights. I also think, given his famous resentment of being upstaged , that more than likely he has encouraged his sons and DIL not to take on many assignments because he is threatened. JMHO.
I'm sure he was spoilt (and still is to a degree) and it came as a shock to him that his first wife was the star in just a few months while he was the royal heir and had worked for the country for a decade already. I'm sure most of the people who were told they are something special from very early on (not to mention being called HRH, Prince, being treated according to their place in the line of succession and see that people bow and curtsy even when they are still a child) react like that. But I think when everyting broke down, his many mistakes became public and he was the villain in the eyes of so many people he did learn a lot about the real life outside the palace and the royal family.
Charles did several mostly military events with Harry in the last years and there were a few occasions when the crowd didn't let go of Harry and Charles had to wait. But he never looked envious or annoyed by that. On the contrary at the opening of the first IG when Harry got a lot of praise he looked very proud. So I don't believe that he doesn't want his sons to overshadow him. But I think that the realtionship with William is very complicated and strained. And while Harry doesn't seem to have a problem to cooperate and support Charles (and Charles supports him in his way) while still doing his own stuff I doubt Will does the same. I think Will was raised to be number one and that everyone is there to support him and he behaves accordingly. JMO
Re sliming down the monarchy: I think this has to be done. Today travelling is much easier and you can do more in less time while still being able to have a family life. And with HM having 4 children and so many cousins the monarchy needs to slim down. The only problem I see is that every member of the working royals really needs to do his/her part to keep the monarchy relevant but the heir to the heir and his wife don't look like they want to.