Hmm. IIRC, there are already designs for winch-free lifts, more akin to trams, that have multiple independent cabs in same shaft...
IIRC, OP's thoughts have been explored by designers of 'city towers'.
Even with active mass-dampers, single 'tapered towers' proved vulnerable to wind sway and, IIRC, tidal forces. The 'jack-up' foundations to counter settlement, uneven or otherwise, are non-trivial. There's safety / evac issues, too...
IIRC, a truly heroic Japanese mega-design met some of these concerns with a toroidal, 'ring donut' arrangement. A ring of w-i-d-e close-coupled hi-rise towers were deep-linked at enough levels for ample housing, prompt horizontal evac and, yes, 'people mover' mini-trams. The tapered towers' flexible couplings provided quake & storm resilience. The 'atrium' held a stack of circular parks / urban farms set at those link-levels. There was enough air-gap at each such level for helos, air-taxis, smoke clearance and partial natural lighting. Sprinklers & fire-monitors on underside of each atrium level formed a useful precaution...
Robust enough for Godzilla to use as cat-gym, the project would require more concrete and rebar than the 'Three Gorges Dam', so no takers yet...