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?