Rare is the speaker that presents a constant load all the way across the audio spectrum. In my experience, speakers tend to dip to lower impedances more than bump to higher ones, respective of the advertised impedances. Most amps are spec'd for what they can drive at a specific frequency, into a specific load impedence, and do it continuously and what they can do for a very brief period. Higher current amps are able to briefly, or continuously, supply more current when needed. Remember that power is not just ExI, It is also I2R. What that means for us, is that the amp is able to handle lower impedances, either briefly or continuously. Some speakers just require amps that can supply the extra current that may be may needed at various places across the spectrum.
Output impedance vs load impedance is important in such regards.
I prefer amps that can do it continuously. Hence they are spec'd for lower speaker impedances from jump street, like the Yamaha M80, or M85, which are spec'd to drive 2 ohm loads all day long. Concept receivers can push more current than normal, but only briefly, but that is still a good thing.
Bottom line: high current amps can be an important consideration in the audio equation.
Enjoy,
Rich P