What exactly is critical listening? Who does it?

RE: "So, my question is why is this a superior mode of listening, what makes it inherently superior? And why is consciousness of sound quality or qualities necessarily a critical mode of listening?"

I don’t think either kind of listening, critical or pure enjoyment, is superior. Each has its place. Critical listening has it all over enjoyment listening if the goal is to describe the sound to others in ways that they can relate to -- but other than that, enjoyment listening is the default mode, at least for me.

As to your 2nd question: purely my opinion, but when "consciousness of sound quality" dominates one’s mind while listening, this allows one to be critical--as in discerning, receptive, paying attention to all aspects of sound. When I’m doing pure enjoyment listening, of course I’m conscious of sound quality as well, but not in a way that dominates my thinking. Rather, my feelings of pleasure and enjoyment of the music dominate my thinking.

I suppose critical & pure enjoyment listening aren’t mutually exclusve: just a balance that shits to critical or enjoyment in a given circumstance. I have experiences with headphones in particular where I cycle from critical to enjoyment listening and back again. For example, tonight I’ll hear the Verite Open headphone via either the Wells Milo headphone amp (SS, very powerful and smooth) or the Icon Audio HP8 amp (transformer-coupled, an entirely different sound). Switching between amps, I always listen critically at first, to pick up key differences in how the amp sounds vs the previous one. But then I relax back into enjoyment mode.

Critical listening and enjoyment listening modes are not mutually exclusive

I should have added they can both occur simultaneously. Listening is a skill, one that can be improved through mindfulness and practice, judgement is where we go wrong, free your mind and your soul will follow. 

I am happily at a place where I am satisfied with my components and vibration control and non-equipment issues like speaker placement only needing to make any decisions except when something breaks or wears out. My cartridge will be 5 years old next month and I listen about 750-1000 hours a year so it should wear out soon. Then I'll have a decision whether to buy a replacement, get the stylus replaced (and anything else that needs bringing back to specs) or possibly upgrade. So whenever I listen closely, it usually is specifically to an album that is one of my favorites, or if I have gotten a better pressing of one of my records.

The best times are when I am not listening closely and notice something I never heard before or if it sounds better  than expected. When it sounds worse it's a bummer.

I know it can always sound better with a better component, but the incremental improvement as this point is not something I want to invest the time or $$ in.