Almost all discussions around space elevators focus on the cable itself, how to manufacture and deploy it, and completely forget about the issues that would arise afterwards:
1) How do you attach the climber to the cable without affecting its structural integrity? By squeezing it really hard? A material that's optimized for longitudinal tension strength is probably not very tolerant of lateral compression.
2) How do you provide power to the climber? A regular electric cable can't support its own weight, so either you have to attach it to the climbing cable, or you have to make it from the same material.
3) Is it even worth it? The climber needs to cover a distance of ~36,000 km, so even at 200 km/h it takes 7.5 days from the bottom to geosynchronous orbit. How many climbers and what payload can the cable support at the same time? Refer to issue #1 regarding limits in speed and mass per climber.
The throughput in tonnes/day is absolutely abysmal in relation to the immense upfront infrastructure cost per elevator. Compare this to SpaceX's Starship, which is getting closer and closer to fully reusable 100 tonnes to orbit in minutes. Space elevators will stay science fiction forever, not because they're infeasible, but because they're useless.