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.
The voltage of the grid will fall MUCH less when you pull current.