Slew Rate?

Repeating something I have mentioned before, "settling time" tends to be the same as rise time, but really is not of interest in audio amps because musical sounds have leading edge transients, but die away gradually. This characteristic is used in the restoration of antique recordings. They are played backwards, and any transients that appear are identified as defects, and removed.

The only way that an amplifier can deliver current is by applying voltage to the load. You cannot deliver amperes without volts, so it is not an either/or proposition. However, an amp with a "weak" power supply might be designed to deliver a high voltage that permits signal peaks without clipping, but which, when applied to the load, will result in current that the amp cannot sustain very long. This is why "music power" ratings can be so much higher than continuous rms power ratings. A "weak" power supply produces high rail voltages when the signal is small, but these voltages drop greatly when a large signal persists, because the transformer and rectifier are too small to keep the capacitors charged up. Larger capacitors can help here. "Weak" power supplies actually do make sense because they are economical, and to tell the truth, music signals are characterized by peak voltage far in excess of that which corresponds to the rms voltage. It can't hurt to have a power supply that is capable of pumping out the amps without strain, but unless the audio signal is loud, compressed and peak-limited (pop music perhaps) this capability will not come into play.

Overshoot on transients has, in times past, been deliberately introduced so as to compensate for the slow rise time of loudspeaker drivers. I don't know if anyone does this anymore. A good idea in theory, but probably difficult to execute without problems.