Why would anyone want class AB amp when class A always sounds better ?

Class A amps (almost) always measure better,

Also not true.  AB can have vanishingly low distortion. Compare for instance Pass' class A designs.  He deliberately designs for euphonic distortion.  He does not make absolutely low distortion amps nor does he wish to.  It's exactly his house sound.

So, neither sound nor measurements are better universally by class.  Class D for instance can best many Class A for distortion and low output impedance. Not always though.

What is true is that the crossover distortion vanishes with Class A.  Sound quality, output impedance, distortion, all vary greatly and overlap with other class of amps. 

Hear, Hear... jw944ts! 
I have, after almost 50 years in this wonderful hobby, ended up with "mostly" tube equipment, push pull A/B, with Martin Logan hybrids and auto formers (take it easy georgehifi, my preference, my system). Sounds great to me (and associates).
My best friend(s) have older Morrison mono class A SS amps, with Morrison (new model) speakers. Sounds great to us.
Our other friend, has Mark Levinson 300w amp and some European (German) stand mounts. Sounds Great.
I think the point here is that, yes, there are general "rule of thumb" guidelines but.... there are no absolutes (class A is superior.... etc). Way too many variables and always an exception to "the rule".
My take on all this... that it is a valuable endeavour  to mix and match, swap equipment in and out, socialize and listen to music (and gear).
As jw said...ENJOY....

Seems to me that everyone is addressing the question solely with regard to SS amps. Ask the same question about tube amps and I think it would be generally agreed  that a pure class A SET will win out over a class A/AB push pull triode as long as they are matched to the right speakers.

J.Chip

Why would anyone marry a brunette when blondes always have more fun?

Develop some awareness of the topic
https://www.electronics-tutorials.ws/amplifier/amplifier-classes.html