TL;DR: super-AI and functional nanotechnology may as well be magic. You can handwave this as you see fit.
This is going to be awkward, but not insurmountable. Tethers of any kind, including non-rotating skyhooks, rotating "rotovators" and fixed space-elevators will be subject to continuous damage from micrometeors, not-so-micro meteors, cosmic radiation, reactive oxygen species in the upper atmosphere, weather... the list goes on. You can't build one before you've got the ability to maintain it. This might not necessarily need nano technology: regular-scale stuff could be done via maintenance robots that coudl crawl up and down the tether, knitting broken strands back together or removing irrepairably damaged strands entirely and extruding new ones into the tether to keep it in one piece. Your tethers will already have to do this.
For non-fixed tethers, eg. your skyhook, you will also have to deal with the problems of orbital decay. You're dragging a big heavy cord through the upper atmosphere, and all that drag is going to slow you down and eventually you'll crash. If the tether was built with parking in mind, it might be possible to rotate it up so it was parallel with the Earth's surface, or perhaps even raise it into a higher orbit which is longer-lived. This might be a sensible way of doing major maintenance on the tether of the sort that can't be done whilst it is still "in flight".
In either case, changing orientation, changing orbit or even just simply maintaining your orbit requires power and reaction mass. You can get power from the sun, but those solar cells don't last forever, especially in the punihsing environment of space, so you need spare parts. Those spare part stockpiles probably won't last a thousand years (because what would be the point of building a stockpile that bit?) so you'll need an in-orbit industrial base, and you'll need a supply of raw materials, probably in the form of rocks flown in from the asteroid belt or lifted off the moon. Those rocks would also provide reaction mass for the manoevering rockets, without which you won't be able to stop the tether's inevitable orbital decay and firey demise.
A suitably super super-AI could probably arrange this complete industrial base for itself, if it didn't already exist, assuming it had access to some initial fabrication facilities and suitable reaction mass. It might also conclude that there are more sensible things for it to do than to wait for the frozen meatbags below to warm up, and it would be fully capable of waking them up itself and retrieving them. As it hasn't done that in your setting, you'd have to wonder why, and rather hope that it was at all interesting in maintaining a human-friendly space infrastructure that it would have no need of itself...