Amplifier current vrs watts; why is current more important?

Current is only of import when speakers dip to low impedance or have challenging phase angles, or both. Otherwise it is of little consequence.

There are so many wrong's in this statement.

Your typical amplifier outputs volts

It may be a bit oversimplification.  Two power amplifiers that have the same power supply voltage may not have the same output current.  Ultimately it's neither the voltage or current, but the overall design that determines the quality of the sound.

You may have a massive 800W power amplifier with massive output current but it may not provide quality music reproduction compared to a more modest 250W amp.

'High current' is a marketing thing. As others have pointed out, you don't need that much current to drive high power into low impedances.
If you are looking closely and doing the math, it becomes apparent that the current 'rating' of many solid state amps isn't real. Here's more on that topic:http://www.atma-sphere.com/Resources/Common_Amplifier_Myths.php
Because those high current ratings you often see aren't about the output of the amp, there are tube amps with similar current ratings, such as our own MA-2, which has a 'current rating' of about 50 amps.