One of my major concerns when looking for a loudspeaker system is it's ability to hit the required volume level(s), across the required frequency range, without compression or excessive distortion.
I've explained it before, but I have a decent amount of experience with PA systems and "live" sound work, and learned with those systems before moving to more home audio stuff. (This next statement is part opinion and part fact) Most typical "hi-fi" style speakers have poor dynamic compression performance, and even at lower levels I feel that I can tell the difference between a dynamically-compromised speaker and one that is capable of significant output.
This could be due to directivity or other effects, but I've also listened to speakers with known good directivity and experienced the effect I am describing. It's the "this sounds real" effect. It's the point where I could actually be convinced that there is actually a vocalist, instrument, etc, in the room, and not just a loudspeaker system playing back a recording.
IIRC, a while back I may have caused a fuss under the blade or LS60 threads where I described this after having heard a pair... They sounded amazing, great frequency response, but it didn't sound real. The JBL HDI-3800 towers, and their other PA systems, on the other hand, have that effect where they sound real. Upon analysis, the main cause appears to be drivers that can actually handle consistent high output levels without compression and distortion. (Not just a quick sweep or burst, but sustained high output)
For the record, I don't listen loud. My current setup rarely exceeds 70 dBC, and even in a large space I generally don't exceed 90 dBC except for a short while. But I have designed systems capable of very high levels (in excess of 110 dBC, a phone mic is inaccurate above that level), and they do get used at those levels on occasion. I've also helped design systems that will probably never get turned up past 80 dBC except for when they want to show it off, but having the horsepower to (when needed) blow the windows out means that it will sound good at 80 dBC all day, and that I have headroom to put in EQ boosts if desired.
So in conclusion, my big concern when selecting a loudspeaker system is dynamic capability, followed by directivity and then frequency response. The other criteria listed are usually reasonable when my main 3 are met. I don't particularly care about how easy something is on an amplifier, amp power is cheap. For bass extension, sub 80 Hz, I just deploy subwoofers, or if I have the headroom an EQ boost.