Why do speakers improve with more powerful amps?

So why the hell do people buy inefficient speakers? 

I raised basically this question.

https://forum.audiogon.com/discussions/low-sensitivity-speakers-what-s-special-about-them

Loudspeaker design nearly always involves trade-offs regarding sensitivity, size, bass extension and dynamics.  If you don't think a loudspeaker the size of a large refrigerator is too big, then you can have all four.  Lo wattage and high wattage systems have their pluses and minuses and either approach can deliver excellent sonics.

I recently asked a very similar question at DIYaudio and got some really in-depth responses which I felt really got to the root of the amplifier/speaker matching problem.

Take a look:

https://www.diyaudio.com/community/threads/if-speaker-impedance-wasnt-complete-what-are-we-missing.398934/

In support of what others have said, such as Carls, I have a 100 watt amp, and I bet I am only using the first 15 watts.  I think where it becomes more confusing is when a passage has huge bass it or huge horn entry, does the amp have the ability output alot of current in a short period of time.  This is where I could notice a difference between an AVR and my AHB2 although both has a similar wattage value.  

It's the quality of the signal powering the speakers that would seem to be very important.  A large reservoir of power well designed within an amplifier allows for the frequency demands to be there when you need them.  Presumably a higher powered amplifier would achieve this better than a lower powered amplifier.

Are there other factors that improve a speakers performance when an amplifier offers more power?? And why is this