i think i would just wrap the carbon nanotube cable with aluminum wire and just use the same kind of induction that maglev trains use. this would allow you to climb the tether without actually touching it, and you could actually achieve a greater speed so you could get to the station faster and wouldn't spend the whole trip twiddling your thumbs. this would also allow you to decouple at velocity relative to the cable so you can actually accelerate along the cable in the absence of atmosphere and have a nice bit of exit velocity when you disconnect. this would save

of delta v when getting to the outer solar system.
still i think the real answer to utilizing space would be orbital fuel manufacturing. surely you can scoop up enough usable material from the solar wind, and process it into fuel and other essential items. surely there is hydrogen, helium, helium 3, deuterium and tritium there could all be gathered and separated. couldnt run chem engines but fusion reactors would love the helium 3, the rest could be used as propellants for electrical engines. scatter such automated stations across the solar system, and the requirement for disposable spacecraft would soon vanish. lunar and asteroid mining would produce the metals needed. linear accelerators would work well for lunar launches.
nanotube fiber composites coupled with hybrid rockets and air-breathing reusable first stage engines would be used for human transport to space. once all this infra structure exits i don't think you'd really need a space elevator. everything you need for space exploration would come from space, only requiring human transportation through the atmo. stop looking at the cost of shuttle missions as how much space costs. the shuttle is a brutally inefficient launch system.
when you look at the problem of deploying the cable, you cant just raise it from the ground, you'd have to lower it from the asteroid we use to anchor it (assuming asteroids are stable enough), which would mean it would have to either be manufactured in space, or lifted in sections. so you see the problem, installing it would require more space infrastructure, yet that infrastructure would so greatly reduce the cost of space exploration that it would be hard to justify the space elevator.