@Arkandel The short answer is "Nothing!", of course, but that's not very helpful.
I'd expect that doing that would give more flavor than even a very carefully crafted offstage setup (assuming that staff are good at playing the Princes). It brings with it the usual issues that accompany having very powerful in-setting characters run as NPCs, and it wouldn't surprise me if Amber players as a group are sensitive to some of those issues due to past history, but just because something has worked out badly in the past doesn't mean it can't work out well in the present.
Removing them from everyday life also runs the risk of suggesting that things that the Princes can't be arsed to deal with aren't particularly important. When they do come out, it's a special occasion, but does that make it less special when they aren't around?
I don't think there's a simple yes or no answer. It's about what you want out of a game. To some people Amber is the Princes and Princesses being awesome and doing the things that make them awesome. To other people it's less about those specific characters and more about the environment they exist in. Do I sound wishy-washy? It's probably 'cause I am, since I feel a little uncomfortable with both ends of the debate.