Speaker sensitivity vs SQ

My first thread at AG.

Millercarbon continues to bleat on about the benefits of high sensitivity speakers in not requiring big amplifier watts.
After all, it's true big amplifiers cost big money.  If there were no other factors, he would of course be quite right.

So there must be other factors.  Why don't all speaker manufacturers build exclusively high sensitivity speakers?
In a simple world it ought to be a no-brainer for them to maximise their sales revenue by appealing to a wider market.

But many don't.  And in their specs most are prepared to over-estimate the sensitivity of their speakers, by up to 3-4dB in many cases, in order to encourage purchasers.  Why do they do it?

There must be a problem.  The one that comes to mind is sound quality.  It may be that high sensitivity speakers have inherently poorer sound quality than low sensitivity speakers.  It may be they are more difficult to engineer for high SQ.  There may be aspects of SQ they don't do well.

So what is it please?