I don't think there would be parts not yet existent in robotics. The elevator itself is pretty much a cabin moving on a rail. It may not be built from off-the-shelf components, but building it shouldn't be difficult.
There are other reasons however, some of which are mentioned in the same Wikipedia article you have linked to. So additional to "not having cables strong enough" and "the slow speed":
Cable wear: The cable used for the elevator will be partially below the clouds (subject to warmth, dry air, rain, snow, wind, tornadoes etc), partially in and above clouds (Humid, Cold), partially in vacuum (extreme cold, no air pressure), sometimes in the sun (extremely hot outside atmosphere), sometimes in the shade (near 0 degrees outside atmosphere) etc. So any cable used for the elevator should be able to resist all these environments.
The situation is made worse by considering the friction between the "wheels" of the elevator and the track, chemical reactions with the environment, radiation, solar flares and even biological reasons (for example an eagle deciding he doesn't like the cable, or a sort of moss evolving to take advantage (aka eat) the material)
- Cable repair: Sooner or later, the cable needs repair. There is no such thing as "indestructible". Now imagine, how would you repair such a long cable? Surely, you can't just cut the bad parts and patch it, because the "welded" parts would probably not be strong enough. Replacing the whole cable would be an option which is quite expensive, given the special tough material and the long length. So, at least this issue would delay building the space elevator until they are sure it's doable.
- Safety of the elevator: One challenge is to be able to provide safety for the elevator. For example, how would you prevent asteroids from damaging it?
- Safety of the passengers: Another challenge is to provide safety for the passengers. With a slow speed, passing through the Allen radiation belts is not healthy. This means that there should be adequate radiation shielding on the elevator, adding weight to it and making it harder to build.
- Safety of others: Yet another issue is providing safety for others, such as airplanes or space satellites. Most probably they would all need to become aware of the existence of this new concept, involving reprogramming a large number of air/space-crafts.
In conclusion, probably the most important thing hindering the construction of the space elevator is the construction of the cable. Not only should it be strong enough to withhold the stress, but it should be practically able to withstand any kind of environment. Other reasons would be minor nuisance in comparison, but still they are and would further delay the deployment of the space elevator.