As you say, the skyhook needs an upper limb, with a long enough cable and/or a massive enough counterweight so that the net force on the section of cable at geosynchronous height is zero. If the cable is long enough, you can go from bottom to top with no net expenditure of energy.
The problems of course are the need for an *extremely* strong cable material, and the enormous capital cost of the system, and of the infrastructure to build the system, and the current lack of need for such a high capacity system. The idea goes back at least to 1960, when Y. N. Artsutanov wrote about it.
The free-floating tethers, either rotating or vertical, do need reboosting, if you want a net flow of mass upward. Their initial advantage over chemical rockets is that you can take your time, so you can use high-efficiency, low-thrust methods like the geomagnetic motor, or ion jets.
Later, if you extend the tether system further up, lunar regolith contains an enormous amount of energy, by virtue of its location at the top of the local gravity well. Lunar material flowing down through the tether system can offset a comparable amount of material going up. Once such a system was in place, you could go to the Moon -- or anywhere else in the Solar System -- for little more energy than you need for a suborbital flight.