Surely the Eyrie would be notionally easier to besiege than Casterly Rock, as there’s only one way up or down, so if you just surround the entrance you’ve cut off access to it?

A couple things: first, to get to that point, you have to force the Bloody Gate, then get past the Gates of the Moon, which is mutually supported by Stone, Snow, Sky, and the Eyrie itself. That’s going to hurt. 

Second, that’s assuming that Stone, Snow, and Sky don’t have their own winches to pick up food deliveries from the valley, or hidden paths/tunnels for their donkey trains to take food up. Which probably means you need to surround the base of the entire mountain.   

But even if you succeeded at that, you now have to feed your army against an opponent with years worth of stores, and as you’re waiting, Houses Hunter, Redfort, Waynwood, Melcom, Royce, and Grafton are mobilize to break your siege, and the Belmores, Corbrays, Lynderlys, Coldwaters, Pryors, Sunderlands, Eleshams, and Waxleys are disembarking in Gulltown.