Personally, I would only class soccer as a 'big-time' international sport out of those 3. The top level of basketball is the NBA - there isn't an international competition that generates huge publicity or global interest. In terms of handball I couldn't name you a single team, let alone the name of any international competition. Just because these sports have loads of international fixtures doesn't mean that their respective international structures are commercially successful.
Basketball is pretty huge over the globe, probably the second most popular team sport. There are professional leagues over the globe and the WC and Olympics are far more prestigious than the NBA. Handball is the second most popular sport in Europe and is also popular in Asia, SA and Africa. I think we underestimate the popularity of these sports in the UK, because they aren't as big, but I wouldn't complain if RL could have the international appeal basketball or handball have.