You forgot to use the word ‘just’.
A tall tower sticking out along the earth’s rotational axis with a n unsupported cable running at right angles from the top all the way to the moon. Even if the top of the tower was always at the same distance away from the anchor point on the moon (which I’m sure can’t be the case), that’s still a heck of a long washing line to string from the top of a vertical pole.
HAH!
Took me a couple of minutes to gt that one.
You’d actually want it to extend slightly past the libration points (to maintain tension on the cable) but this is essentially true; owing to tidal locking and resultant slow rotation of the moon, lunasynchronous orbit is very high, and although the tensile loads will be smaller (but still well in excess of current material science state of the art) you’ve going to have even more cable to manufacture.
There’s another problem as well (besides the fact that you’re putting your terminus at a point where dust and debris tend to collect): there’s no way you can use the Moon’s rotation to “launch” stuff out of orbit (or toward the Earth). It’s true that there’s a lot of rotational inertia in an object orbiting the Moon at that distance, but it’s all cancelled out by the gravitational attraction of the Earth; this is, in fact, why there’s a libration point there in the first place. Sure, with a little nudge you can send something back down into Earth’s gravity field and perform a swing-by maneuver there, but that’s not all that impressive, and by itself won’t even get you out of Earth’s sphere of influence.
Ixnay on the Moonstalk. It’s less useful than a ballpeen hammer in a plumbing emergency.
Stranger
Well, how about an orbiting continuous equatorial Lunar ring-station, like in the fim version of Starship Troopers? 
For what purpose? The Moon, despite the appeal (I mean, we can actually see it in the sky) is essentially a dead end. There’s little there that can’t be more readily had from a Near Earth Asteroid more cheaply. I can’t think of a single valid economic argument for colonizing the Moon (food production? Seriously, what was Heinlein thinking?) and very few technical or scientific arguments that even stand on their own merits much less work to justify otherwise loss-leading activities. Putting footprints on another celestial body was a benchmark, a goal to achieve, and there were modest scientific reasons for doing so (confirming the origin of the Moon) but beyond that it’s just not a very useful body. If I were designing a long term space development (as opposed to flag-waving exploration) program, I’d focus on the asteroid belt and/or the Jovian system as being most worthwhile in terms of both industrial and scientific value. The Moon is just a test article for the real thing.
Stranger