Space elevator can only possibly work if it is anchored at the equator because it’s motion must coincide with the plane of earth’s rotation. Otherwise it will wobble back and forth as the earth rotates, the magnitude of the wobble proportional to the distance the anchor is from the equator. Anchor it more than a few degrees from the equator it will probably either crash down to earth after a few days or rip itself from the mooring,
Also, the notion that space elevator, once completed, can dramatically lower the energy cost of sending mass into orbit is scientifically unsound. It doesn’t matter how something gets into orbit, it has to gain both the potential energy of its new elevation, as well as the kinetic energy, and angular momentum, of it new orbital velocity. If you just send it up in a space elevator, all that energy and angular moment the payload must gain to go up, has to be taken from the space elevator. The space elevator does not have an infinite supply of either. If, over the long run, more mass goes one way than the other up or down the space elevator, either the entire elevator will lose enough energy and angular moment to the new payloads to collapse because more mass went up then came down. Or the elevator will gain so much energy and angular momentum from its payload because more mass came down then went up, it will rip itself from its mooring.
So the space elevator has to burn fuel to adjust its own energy and angular moment as payload goes up and down.