Sigh.
The whole point of a space elevator is that the centre of mass of the cable is at geosynch orbit (well slightly past it). There is no need to continuously thrust to hold the cable up because the rotational speed of the planet will fling away the cable.
The reason that the cable stays up is the gravity drops off by the square of the distance but distance travelled by the cable per hour at any height increases. The period of rotation is always 24 hours (give or take some lean) but the circumference described is greater as the height increases.
What this means is that at geosynchronous orbit the force downward due to gravity (at that distance from earth) exactly matches the centripetal forces from orbiting the planet (or more accurately attempting to fly off in a straight line but the gravity of the planet curving that line).
Below geosynchronous orbit if you are orbiting the earth once a day you're doomed to crash into the planet without active energy input (the rate of curvature is higher than your speed thus you'll hit), above geosynchronous orbit if you're still orbiting the earth once a day you're doomed to escape the planets' gravity altogether and go flying off as you spiral out. This is why low orbit satellites orbit so quickly and high orbit satellites orbit so slowly, the relative strengths of gravity at the different orbits, plus the distance travelled to complete an orbit dictate the speeds.
The parts of the cable that are above geosynchronous orbit are attempting to escape earth, and the parts of the cable that are below geosynchronous orbit are trying to crash down to earth - the reason the cable stays up is that these are balanced. The entire weight of the cable (less the reduction due to centripetal forces) below geosynch (and adding the tension in the cable necessary to lift a weight) is indeed all passed through the portion of the cable in geosynch orbit (it's actually a huge section as 1 metre from geosynch either way isn't much difference). But the cable isn't a cable, at least not in current designs, it's a ribbon varying in width from thinnest at the ends to widest at geosynch as the loads vary - there are also thickness variations due to expected damage to the cable, ie at certain orbital levels micro-meteorite (and space junk more likely) impacts are very likely so the cable would be made wider to compensate.
Please note that this doesn't allow anything to be pulled up the cable, it is just supporting itself, what you need to do is stick a nice big weight on the end of the cable on earth, like say the base station weighing in at hundreds of tonnes (but not supported by the cable, supported by the earth's surface) and then extend the cable upwards until it has a tension high enough to do something useful. The tension in the base of the cable where it meets the base station will be at least the weight of anything you want to send up the cable (and have more because acceleration increases the force).
The real problem with understanding this is that humans live on such a small height variation and deal with speeds so slow that you cannot easily imagine what happens to gravity over those distances and how much energy is involved in those speeds - if you accept that orbital mechanics isn't the same as you running about on earth it becomes much easier to understand.
No revisions of any laws (other then potentially materials sciences as we don't yet have a material that has a strong enough tensile strength but low enough weight) are required for this to work, all you need to do is understand them.
The 'energy' you use to raise the payload is electrical converted to kinetic converted to gravitational potential. If there happened to be a mountain of a magical material that reached out into space then climbing that mountain would not violate any laws of thermodynamics.
Towers have problems, compressive strength vs weight, mechanical strength of the earths crust, and major issues with stability. Compression is a positive runaway scenario (the tower bends slightly and the weight causes the bend to accelerate), tension has a negative (or neutral I suppose) runaway scenario (the cable bends and the cable tries to straighten out).
Z.