Amplifier current vrs watts; why is current more important?

The challenge is the fact amplifier manufacturers don't always post that info. Then it's back to looking up reviews, forum posts, etc. 

Even the size of the power supply isn't listed often, much less current delivery ability!

I'm currently planning an amplifier upgrade (current one is an Arcam PA240), and read through this thread with a lot of interest as my speakers (Focal Kanta 3) have some pretty large impedence dips in lower frequencies which has led me to look for an amp that has strong current capabilities.

I'm far from an electrical expert but I'm starting to understand the above. So far the criteria I've gathered from research is to look for:

• Amps that double wattage from 8-4-2ohms
• Amps that are 2ohm stable
• Larger power supplies

Is there anything I'm missing in terms of specs to look for that indicate good current output? Amp manufacturers don't make this easy as many list out spec sheets with varying degrees of information.

Thank you for the detailed response, I'm still learning about all the ins and outs of this topic but think I've gotten a decent grasp on it now. And I totally agree with your point on sound, I've never been a 'spec buyer' but like to balance good objective performance and subjective performance.

While my Arcam PA240 is ok at higher volumes, it lacks low-mid volume punch particularly in the bass department. And I want to get a better quality amp anyway as my system is at a much higher level than when I first got the Arcam.

For what it's worth, the amps I've shortlisted are Pass Labs x250.8, Parasound JC5, Coda 8, and Musical Fidelity M8 series. Also have Krell on the radar.