Just had a look at the Pi's schematics and I see the main regulator is a ncp1117 (3.3v).
Having a look at it's datasheet, it quotes the drop out voltage (@800mA) as:
Vin – Vout = 1.1V
This means that 4.4V is needed as a minimum input for the device to maintain regulation.
Now 4.4V is only 0.6V below the rated 5V recommended input and it's very likely that the cheap power supplies are dipping below this and causing the regulator to drop out of it's regulation range and causing the numerous problems.
Just a thought.
Perhaps one of the designers could confirm/dispel/add to this.
Just noticed that the 5v supply also goes to the BCM2835 where it then generates some other needed voltages.
Not knowing the Broadcom chip set, perhaps it has a built-in voltage supervisory circuit which holds the processor in reset until the supply (5V) is up to spec?
Again the lower supply given out by the cheaper supplies are causing the problem.