Plus its not like you're just going to be able to move this flimsy cable
out of the way, unless you put thrusters in the middle or something.

Well, at http://www.isr.us/Downloads/niac_pdf...0.html#objects
they propose moving the space elevator by moving the anchor point
(which they have on a Sea Launch style floating platform, largely so
that they can move it).

In fact, that site (in particular
http://www.isr.us/Downloads/niac_pdf/contents.html and in general
http://www.isr.us/SEHome.asp?m=1 ) makes a pretty good stab at
answering the obvious objections to space elevators. They answer the
usual questions of the form "oh but it will never work because of X"
for a variety of values of X.