Because a real space elevator would go all the way to geostationary orbit where the orbital speed is the same as the rotation of the planet. Remember that the further up you go, the slower the orbital velocity, i.e. the "horizontal part" shrinks until the "vertical part" becomes the whole thing.

In fact, the way to build a space elevator is not to build a "tower to space", but to put an anchor rock into geostationary orbit and then hang a cable down to earth from it. You can then connect the cable to terra firma so that it doesn't sway, but in terms of the main forces involved it's hanging down, not standing up.