There's not been that much progress in the 20 years since Brad Edwards proposed the first actually feasible space elevator design. It's difficult to say whether ultra-strong CNT cables are something that's perpetually 20 years away, or something that follows the "progress is slow until it suddenly isn't slow" exponential curve.

Would be cool, but as far as I understand, even long, continuous carbon nanotubes wouldn't be strong enough to carry their own weight.

Not if the cable has constant cross-sectional area, but actual designs call for a cable with an exponential taper, and that would make a carbon nanotube cable feasible. With currently available engineering materials the taper ratio would be too large (~10^8 for Kevlar, ~10^33 for steel…)

And I thought the cable had to be made of spider silk hadn’t?

Too bad someone will write an article titled "Will we get space elevators sooner than we think?" and Betteridge's law will prevent it from happening.