Should the ICC have universal, full compulsory jurisdiction?

[Kingdom of Grolsch]

Having interned at a UN court now for almost six months - give or take a few weeks - I've had the privilege to experience first-hand what the international judicial system can and cannot realistically achieve. One issue in particular has caught my attention. It seems that some sovereigns are avoiding full responsibility for their actions in war by not recognizing the International Criminal Court's jurisdiction or the International Court of Justice's compulsory jurisdiction. The ICC, given  what it deals with, is especially interesting in this context. The states concerned can roughly be divided in two groups: less-developed countries with a relatively high number of government officials who where involved in the gravest crimes, and a group of countries which traditionally either are involved more than average in wars or do not tend to subject themselves to external, international oversight easily (with the United States and China being the most notable examples).  It seems somewhat challenging to have an effective global judicial system for prosecuting the gravest crimes when not all sovereign nations recognize the Court's jurisdiction.

My question to you is: should the International Criminal Court (Wikipedia | Official Website) be able to prosecute people for genocide/war crimes/crimes against humanity/crimes of aggression, regardless of nationality and regardless of where the (alleged) crime(s) were committed? So, in essence, should it have universal, full compulsory jurisdiction?