... 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.
The inverter will have noise of its own so it comes down to the lesser of two evils for noise. Even in the suburbs where I live, the data is telling me my Giandel inverter has slightly better AC Total Harmonic Distortion than the wall AC. I use to own a power conditioner that showed AC THD.
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.
Those substations are not a few feet away from the audio system. Any power reservoir from the susbstation needs to travel many feet and it will have to go through transformers and the house wiring and electrical outlets likely rated for 15A. The audio system is not seeing anything close to what the substation can offer for power. But in my case with the Giandel 5000W and using the High Current terminals, my ceiling is not limited to 15A. Doing the math, 5000W / 120V = 41.67A. Now, max power is not really what I'm going for but it's beneficial to have wires and circuits that were designed to handle that kind of power. This can come into play during moments of sudden and quick power draws. I once had Kii Three active speakers hooked up to the battery/inverter system. They were not class A monoblocks but not once did I feel the power was lacking when I turned up the volume.
... 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."
I didn't use the word "require". I generally believe if dirt goes in, then dirt comes out. If a clean AC sinewave goes into a power supply, then clean DC power comes out. If a jagged AC sinewave goes into a power supply, the DC signal out would likely have artifacts(noise) from the jaggedness of the AC sinewave.
I have an academic background in EE. I have a BS and MS in Computer Engineering. I don't share this to say I know everything about electricity because I don't. I share this because the decisions I've made regarding how I deal with power in my audio system is helped from this EE knowledge or whatever is left of it. I don't do EE for a living. But above all, I try the gear and nothing beats actual experience. Theory is nice but actually applying theory is better.