Only the first flaw is necessary kill the plan, but I'll elaborate:

1. There is no material with sufficient strength and other properties to construct a space elevator tether.
2. It would get in the way of the hundreds of other orbiting satellites.
3. A single micrometeorite would sever it. There are millions of those.
4. Lightning would shred and burn it below a certain altitude, probably 10-20 km.
5. It would take over 2 weeks to climb, even at 60 mph vertical speed.
6. A modest weight vehicle would stretch the tether hundreds of kilometers. The vehicle would not climb until the tether stress-area product matched the vehicle weight. The ground station would have to take up the slack or spill the loose tether all over the ground. The same problem would occur regardless of whether a vehicle is climbing or descending the tether.
7. A climber malfunction would be a disaster, depending how high it is. It's weight would also impede any effort by the tether nodes to thrust out of the way of orbiting satellites. Like a lead weight on a fishing line.
8. Thermal expansion due to ambient sunlight would affect the tether tension, height and drift of the counterweight.
9. The tether material may be decomposed by ultraviolet sunlight, as are many materials.
10. The tether material may fail in the atmosphere, subjected to moisture, ionizing radiation, oxygen, ozone, hydrogen, hot, cold, high winds and cyclic stress from vibrations.
11. Sections of tether would have to be removed and replaced. How do you support a climber while perfoming this operation?
12. A failed tether would be a hazard as it fell back to Earth. Whatever portion remains in space would be hurled into a higher, probably elliptical orbit by the momentum of the counterweight. The tether would become a tangled elastic mass, whipping around dangerously.
13. It wouldn't place objects in orbits below geosynchronous, since most of the energy of orbiting satellites is horizontal. Access to only one orbit - the lack of flexibility - makes it useless compared to rockets.
13. Even if it were feasible, it would probably be uneconomical. So why bother.

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That was fun. This must be how it feels to be a critic. Always right! Cool feeling.