Can you help me resolve this paradox for a geodesic orbit? Does it maximize proper time to sit still rather than orbit for 1 year? The usual simple statement of geodesics is that when moving from point $(x_o,y_o,z_o,t_o)$ to point $(x_1,y_1,z_1,t_1)$ the object will follow the path of greatest proper time experience, i.e., a clock traveling the geodesic will record more seconds than one going another path.  I believe that is true even if you allow rocket thrusters or other energy addition devices.
But what if you consider a circular orbit which, after 1 year, results in going from $(0,0,0,0)$ to $(0,0,0,1)$?  Wouldn't the proper time be maximized by simply sitting still, i.e., traveling only through time and not space?  The orbiting clock would presumably run slower just as the orbiting clock in the international space station runs slower than one on Earth.  In fact, if you want a physical model, assume a clock on a 400 km tall flag pole synchronized with the ISS during a pass-by (the ISS orbits at 400 km above the planet surface).  One orbit later, wouldn't the ISS clock record less time than the flag pole clock?
Is that a violation of the concept "orbits are geodesics, for which the shortest distance between two 4D points is the path of greatest time"?
 A: 
One orbit later, wouldn't the ISS clock record less time than the flag pole clock?

Yes. The orbiting clock will record less time than a hovering clock.

Is that a violation of the concept "orbits are geodesics, for which the shortest distance between two 4D points is the path of greatest time"?

No, it is not a violation. In curved spacetime there can be multiple geodesic paths connecting a given pair of events. Geodesics extremize the proper time, but that is a local principle. That means that a geodesic has more proper time than any other “nearby” path, specifically any path that is only infinitesimally different.
The hovering path is not a geodesic and is not a local maximum. If instead of hovering the observer goes slightly up at the beginning and then back down at the end then there will less time dilation and a longer total elapsed proper time.
In contrast, for the circular orbit if you make any small deviation you will reduce the total elapsed proper time. So even though the hovering path is a longer path it is not a geodesic, and even though the circular orbit is a geodesic it is not the longest path. And that doesn’t violate the fact that a geodesic extremizes the proper time because they are not “nearby” paths.
