I suspect this is all a bit backwards; the constraint on building height was not so much the lack of elevators as the structural methods of making them very tall. I could be wrong about this as tall towers, say for Gothic cathedrals or towers like the one at Pisa existed close to a thousand years ago. But I gather all of these were major projects that took centuries to complete and sometimes collapsed in mid-construction; you could hardly have every apartment building thrown up in a hurry along their lines, and if you did a lot of them would fall down. (Fewer and fewer as they got empirically better at it to be sure). My impression is, it was only with the development of relatively cheap steels that could be churned out in quantity that really tall buildings became possible.
At which point, steel cable for the elevators is available.
Of course another limit on elevators is that they had better be fast and reliable, if one elevator is the only practical way to reach a high floor. If instead of a single car per shaft, you had a continual ladder like a vertical escalator--I know they had some things like that in some buildings, there's even a special name for them but I forget what it is

--then the thing could be relatively slow and of course if no one has to stop it and start it, it's relatively simple. Probably a lot heavier actually than a single elevator car, however elaborate, and of course if someone falls off they'd just keep falling, unless the "rungs" fill the shaft--in which case there is danger of getting fingers or limbs cut off. And of course it could still break. But something like that, with rungs or even platforms going up continually on one side and down continually on the other, is I suppose the logical ancestor of the elevator.