Almarg...I don't agree with your description of the power supply as "weak". It may have been designed that way.
I like to think about this issue in terms of voltage. The amplifier delivers an electrical output which is the same shape as the input signal, but at higher voltage. When the output voltage is applied to a load (speaker) current will be drawn, and if the amp can't deliver enough current the voltage will fall. This limits amp power. Amp power may be limited by either the voltage it can output, or the current it can supply. It is better to be limited by current because music typically includes voltage peaks of such short duration that the power supply can provide the necessary current for a brief interval by drawing from its capacitance.
The "continuous rms power" by which audio amps are rated is really unsuitable for audio. The "headroom" measurement is important.
I like to think about this issue in terms of voltage. The amplifier delivers an electrical output which is the same shape as the input signal, but at higher voltage. When the output voltage is applied to a load (speaker) current will be drawn, and if the amp can't deliver enough current the voltage will fall. This limits amp power. Amp power may be limited by either the voltage it can output, or the current it can supply. It is better to be limited by current because music typically includes voltage peaks of such short duration that the power supply can provide the necessary current for a brief interval by drawing from its capacitance.
The "continuous rms power" by which audio amps are rated is really unsuitable for audio. The "headroom" measurement is important.