It's not a new idea, really. I remember seeing Nagra come out with a battery-powered preamp for home stereos, many years ago (and I think they still make them).
I also remember hearing back in the day of some audiophiles powering their systems with regenerating UPS from APC (like the kind you'd see in a data center), though PS Audio has been making such units marketed for audiophiles for a while now too.
DC to AC converters in common inverters (where your battery power gets turned into AC wall current) are usually digital (mosfets usually) and unless they're running at 96khz+, which is probably not the case even with "pure sine wave" inverters¹, they could also introduce some noise. Additionally, if you were charging the batteries at the same time, that charger could be carrying some noise into the DC current.
I'd love to see somebody sell an old-fashioned motor-generator set for generating absolutely pure sine-wave current from a noisy input. Give it a really heavy flywheel and maybe some complicated flexible interconnect between motor and generator, to remove any effects of the input power. Or, even better, have the motor crank up some sort of mechanical/hydraulic/pneumatic power storage, then switch off to let the generator run for an hour or so.
Yeah I mentioned it in my other comment, spinning flywheel generator with a motor on one side and generator on the other would work well here too filter out any noise.
Some datacenters have these attached to their generators to avoid the need for huge battery banks. They can actually support several megawatts. They spin constantly at the motor's rpm with a clutch that engaged when the input power is lost. This instantly jump starts the generator, at the grid's phase, and prevents
"Battery" is a sort of nebulous term for a collection of similar objects working in concert, whether that be artillery guns or electrical cells. Consumer electrical batteries are a collection of electrical cells.
Batteries have significant and unavoidable source impedance. When the current goes up and down, as it does once per cycle, tens of thousands of times per second, the voltage of the power bus would move up and down slightly. You can obviously regulate it, but the whole point of what these guys want is an absolutely clean input.
The voltage of the grid will fall MUCH less when you pull current.
It does depend on the load. The grid voltage is transformed to a lower suitable voltage. The transformer has an inner resistance as a battery has.
The AC voltage is rectified and smoothed with a relatively big capacitor. There is a ripple in the DC voltage because the AC can only charge the capacitor in the part of the AC cycle when the varying voltage is higher than the voltage level on the capacitor. If you charge an equal size capacitor from a big battery there will be ripple due to the inner resistance of the battery, but much less ripple because the battery voltage is always higher than the capacitor voltage.
Other things equal the battery will give a smoother DC to the amplifier.
Anyway ripple is not the problem in question here. The problem is various noise from the grid finding its way through the transformer and further into the amplifier. There is very little noise from a battery compared to the grid.
