... a few key things that a powerful battery/inverter system offers over the average 15A wall outlet. It’s separated from what can be a noisy grid ...
And replaces it with a potentially noisy inverter.
It can offer more dynamic power (from a large bank of capacitors?) ...
Greater than the dynamic power available from your local electric utility, which uses banks of capacitors at the substation? I don’t think so.
... it can offer a better AC sinewave (better THD) to power supplies that were designed with perfect AC sinewaves in mind.
A good power supply doesn’t require "perfect AC sinewaves."