My generator is air-cooled, but I can't imagine your radiator is any different from any other radiator...maybe a little smaller, but no different in design. That little tube you refer to sounds like the line to an overflow tank. When the pressure inside the radiator reaches a certain point, the main cap as a pressure regulator will open and allow the flow of coolant to the overflow tank. If the cap is off, there's nothing to prevent the flow of coolant through the overflow tank, so most likely what you're seeing is that flow.
In this case, I'm not sure if your gen-set has an overflow tank since it's much smaller than a typical automotive engine, but I suspect the principle is the same using the tubes as the overflow compartment. In any case, with the cap off and the engine running, you should see the coolant flowing/circulating... if not, you've got a problem... there's just not enough room in that little radiator to let you watch without spilling some....it's gonna take the path of least resistance...and that's the big hole you opened up....LOL!