Technology

Energy density has nothing to do with this.

No it absolutely does, and it matters because:

It’s the cost of how much pollution refining the rare earths and making batteries produces vs the amount of pollution associated with construction of a building with pulleys that move weights up and down.

If an entire building could be supplied with a few elevator shafts and some weights, because the energy density of the system is so high, it would be silly not to do so. But the energy density of these systems isn't remotely close to that. Where as yes, a building absolutely can be built with batteries as a part of it to support its typical duty cycle.

Gravity is just not a particular energy dense form of storage. Its not really debatable. And like you said, building buildings comes with all kinds of other forms of pollution. Not to mention, we could be building them to house people, not pulleys and weights.

Its an idea that sounds good, but once you engage with it seriously, its limits become obvious. Pumped hydro will almost always make more sense. A big tank at the top and a big tank in the basement, and bam. Battery built. not to mention you've got a semi-permanent back up reservoir now built, which could help with flood control, drought tolerance, fire control, all kinds of other things. And you don't need to build new buildings for this. They can go into/ on existing buildings.