And these systems are way larger, more dangerous (wind) and not as efficient as magnetic bearing flywheels.
This is basically the same as pumped water storage, but instead of immense amount of water you can pump between very large reservoirs, you’re limited to one solid weight that’s too light to store nontrivial amount of energy, but heavy enough to be significant problem from engineering perspective. It just doesn’t make practical sense.
It's still basically piss-stakes: the largest wind turbine right now (14MWe Haliade-X) has 765t sitting on top of a 150m tower, so let's say you keep the 150m tower, rejigger it so it can move a 765t weight up and down the shaft without crumpling, and add your weight (a lead weight 4m in diameter and 5.4m tall), how much energy does that store?
250kWh, 2.5 teslas, or a medium-large electric bus battery.
And it's not like you can add that to a working turbine: the tower was set up for 765t not 1500, and you need space for the crane.
A stick of butter and an 2-tonne truck at 216 km/h both "contain" about 1 kilowatt-hour of energy.
The butter could maybe burn your house down but the truck could easily kill a couple of dozen unprotected humans in a fraction of a second.