I was just using car stereo caps as an easy-to-reference bound. (I'm not in tune with current cost of bulk energy storage caps)
As I was saying before, the main difference from your amplifier was the amount of power involved. In general, audio amplifiers have large capacitors on the power rails to smooth out the ripple voltage, and will coast for quite some time as they don't actually need that much power for typical music and volumes.
Here's a similar example - high current USB charging bricks, with a power indicator LED. If you have no devices connected and unplug it from the wall, the LED will stay on for quite some time. The LED just doesn't draw that much current (say 10mA) compared to the capacitance, which has been designed for smoothing the full charging current of 10A. Also the LED will happily ride the voltage down from 5V to its junction voltage of 2.5V or whatever, using even less current. And due to the logarithmic nature of your eyes (ears), you don't fully perceive that dimming.
The caps were way, way excessive for the application, I was just having fun putting them in. And they didn't just smooth out the power, they were capable of running the amplifier for a few seconds. (It was a 125 watt Universal Tiger amp.)
I do know that caps decay over time, but have no idea how that compares with batteries. I expect, though, that long life caps could be built.
I also wonder about using inductors instead. They are, after all, just a coil of wire.
I feel like you're asking the equivalent of why we use DRAM instead of SRAM, or spinning rust instead of flash. And I don't want to shout that down because, as in the latter, constraints do change. But the current state of things is fundamentally based on a cost tradeoff.
Perhaps in the future we'll have supercooled inductors providing power storage, and not requiring periodic changing the way say batteries do. Alas, it's just not right now.
As an aside, it's amazing how little power it actually takes to produce worthwhile sound out of modern speakers. I once wanted to test some tower speakers without a proper receiver around, so I wired up a breadboard with some stupid opamps around poorly-heatsunk transistors, powered by a lab supply. The speakers turned out to be disappointing, but not because of the amplifier!
As I was saying before, the main difference from your amplifier was the amount of power involved. In general, audio amplifiers have large capacitors on the power rails to smooth out the ripple voltage, and will coast for quite some time as they don't actually need that much power for typical music and volumes.
Here's a similar example - high current USB charging bricks, with a power indicator LED. If you have no devices connected and unplug it from the wall, the LED will stay on for quite some time. The LED just doesn't draw that much current (say 10mA) compared to the capacitance, which has been designed for smoothing the full charging current of 10A. Also the LED will happily ride the voltage down from 5V to its junction voltage of 2.5V or whatever, using even less current. And due to the logarithmic nature of your eyes (ears), you don't fully perceive that dimming.