Beck, I had a similar situation with a machine. I disconnected the upper radiator hose from the radiator and put a plug in the radiator where the hose would be. Filled the system with water and fired the engine up. With no thermostat, the action of the pump should push water out that top hose. Only takes a second to check the action of the pump that way.
In my case, the problem was a simple one, the radiator was severely plugged externally. I've had other ones where the radiator was so severely plugged internally that water could not pass from the top to the bottom of the radiator fast enough and thus, it exits the overflow.