Belgian neutrality/sovereignty is a very real piece of European diplomacy and also Western diplomatic thought. Violation of it was one the major sparks of British involvement in WW1. Switzerland is also a neutral nation historically but is Geneva is more of a focus for the global community. Also, as someone else stated, Belgium is a geographical focal point for the major Western post-WW2 European powers without showing favoritism to any specific one or offending Charles de Gaulle.

I know it's because of the Treaty of Brussels, but why was it signed there, specifically?

Unified Europe started with Charlemagne and the Holy Roman Empire whose capitol was in Aachen which is close to Brussels. Brussels is more significant nowadays than Aachen. It is also in the middle between France and Germany so it's a central location.