A diplomacy always needs someone the people can get excited about. If the people don't have someone they can cheer for or look up to, then the people will likely be confused and more worried about what the Government may do. The President is powerful, but his position is that of one for celebration.
People love celebrities, and celebrities are the richest and most well-known people. There really is no better candidate for a diplomatic figurehead than a well-known celebrity. Some will find this ridiculous, of course. But at the very least it bridges quite a bit of a cultural gap between people. That's what you need.
It takes quality people to be diplomatic and make a difference when venturing into uncharted territory. People who have a distinct interest in international affairs by building and mending relationships can be very diplomatic. A celebrity taking this sort of role may be appealing, but overall being affective to the cause is still the most important aspect of diplomacy.
I think that sometimes a famous or celebrity figurehead can help diplomacy. I am not exactly sure if Dennis Rodman going over to North Korea and trashing hotel rooms managed to accomplish anything postive. For all we know, he didn't even try to negotiate with them over any human rights issues or anything important. I think there are normal diplomats who have proper training who can do better than celebrities who are going on their own missions.
No, celebrities do not need to get involved in politics, because they do not have a strong political background, and do not have a lot of knowledge about politics. The way it is going now is good, and only politicians need to be the ones who are spokesmen for diplomacy.