It's called a Beanstalk
It's been a popular SF idea for years, and you zeroed in on the one problem, material strength. We just don't have anything with the right strength and weight characteristics right now. As for what happens if a plane flies into the elevator, it's hard to imagine the size and strength required. I wouldn't want to be in what was left of the plane, and the beanstalk would be pretty much unscathed. As for the fuel burning, yes, nanotubes are carbon and could cook off, but you won't see a 9-11. There are too many ways to dissipate the heat.
A functional beanstalk would make a lot of things possible and aid in raising the quality of life down on the planet.
