is it really easier and cheaper to store the energy needed for a home in a chemical battery?

Yes. A 5kwh battery is about 50kg and smaller than a carry-on suitcase. String 6 of them together and you've got 30 kWh stored with no moving parts. Anker has that for about $15,000, maybe $30k installed.

How much does a 3-story elevator cost? What about one that can capture the stored potential energy on the way down, and not break down?

Actually I was off by a factor of 1000. That Camry needs to be raised to 7.3 km. Or you need 1000 of them. Or some combination of increased weight and height.

And not just that, but safety and cheapness akd accessibility of the materials. Water is pretty cheap and common. If it spills, probably no big deal (flooding notwithstanding) unlike battery acid. Not as likely to explode (sometimes steam explodes stuff).

Hmm… this might be easier to do with an electric car. Put it on an inclined track, and then drive uphill to store energy, and go downhill to release the energy.