I don't think the tether can be made stiff enough to keep the cork screw shape. Geosynchronous is perhaps optional for a high tower near the South Pole. There are some tall mountains to help with the height. The non-corkscrew tether can extend horizontally from the top of the tower for about 50,000 kilometers, which puts the counterweight high over the Arctic circle (not the Antarctic circle) The tether will sag some while the climbers are moving North so the tower needs to be very tall to get above the atmosphere after about 2000 kilometers travel on the tether. If the counter weight circles Earth several times per day, launch near the counterweight, can take the space craft anywhere in the solar system in a few years or less due to the very high speed. Advantages are low laser energy is needed to propel the climbers and none after about 2000 kilometers. Disadvantages are: 1 very difficult to deploy 2 the rotation rate will slow due to air resistance and the moving of the climbers, so we will need to speed up the counter weight frequently or continuously. 3 the specs for the CNT need to be better than that required to build a practical Edwards type space elevator. 4 The South Pole is not a convenient location to launch climbers. 5 I forgot 6 The release speed (near the counterweight) is too fast for many destinations such as GEO orbit, but we can release any where after the climber leaves the atmosphere to get optimum speed and direction of the throw. With excellent CNT specs, if the ribbon breaks, the wide, but very thin ribbon slows quickly in the atmosphere to the local wind speed, and is thus a hazard only to penguins and any aircraft flying over Antarctica. If the ribbon breaks high in Earth's atmosphere the upper portion goes into solar orbit, and will burn up in the upper atmosphere in the unlikely case of re-entry.
A similar design is likely workable most anywhere on a Moon's surface, but much easier due to the lack of atmosphere and the low gravity. Neil