The dedicated 20A circuit is intended to provide power to alot of devices. If you look into the wiring of your house, you will find that a 20A breaker will run power to a whole room or even multiple rooms depending on the number of outlets in each room. The individual normal devices (i.e. not a refrigerator or A/C unit) taking power from that circuit actually draw much less than 20A.
For example, from my rack:
DAT machine rated max draw is 21W
CR recorder rated max draw is 27W
Multiformat player (energy star) rated max draw is 14W
can't find the rating on my preamp but it takes a .5 amp fuse
So if I power on all this stuff, I am consuming at most 1A.
A power amp is different. If you have a 240W/ch amplifier, that is a rated maximum of 4Amp if you are driving it to the extreme. Ususally it draws a lot less continuous current in most applications. The reason that you want a much larger capacity circuit for the power amp is that it may have instantaneous power requirements that exceed the average. The 240W/ch amp may dissipate on average 120W, but have instantaneous power requirements of 8A to 10A when there are loud transients in the material.
The primary reason for having a dedicated circuit for audio equipment is to isolate the audio equipment on a leg that is as far away from other devices as possible, because all those other deveces (lights, TVs, ceiling fans, refridgerators, hair dryers) all put noise back onto the AC line where they are tapped. That A/C line noise gets through the power supply filters and degrades the audio signal.
The reason to put an amp on it's own circuit is that when the instantaneous current becomes high there is a voltage drop on the circuit and if your source devices are on the same circuit, they have to suffer that voltage drop. It's not unlike turning on the hair dryer and seeing the bathroom lights dim. A 1400W hair dryer draws ~13A like a big SS applifier will when it's trying to refill its stores after a large transient on the output. Also the draw of the source components, however minimal, cuts down on the total capacity available to the amp when it really needs it.
You can run wire rated for 20A in your power cords and it won't hurt anything. But if your device draws only .3A, then running wire suitable for 20A is overkill. Especially for source devices that do not have large instantaneous power requirements. If you have wire suitable for 10x the max draw of any device, you're not going to see any benefit from investing in bigger wire. You'll get better returns from investing in wire/shielding topology.