When opening a door, the farther from its axle you push, the easier it is to open because you have more torque, now say at the far edge of the door (the edge that's not connected) there was a handle perpendicular to the door (say a couple of cm's long), and on that handle there was another one parallel to the door and extending from the far edge of the door to almost its axle and only connected on the previous short handle (so kind of a very long door handle). Now when pushing from the end of the long handle would you have the same torque as pushing from the edge, or as pushing from near the axle.
as pushing from near the axle.
The handle and door are a system acted on by an external force near the axle.
As the rod and door are assumed to be rigid the situation is no different to having the space between the rod and the door filled with a rigid material ie having a much thicker door.