What makes you build a system around an amplifier?

Obviously the reason would be one cares more about the amplifier than the speakers for whatever reason. A totally subjective decision.

In the end the best system is the one you like best for whatever reason.

I did it once.  The Cary 300B pure class A SET amp was the reason, as with others here.  Bought when Denis Had was still running things.  Paired with ProAc Response 2.5s.  Dennis H was reliably reported to voice his amps with ProAcs back then.  The Cary came before the speakers.

I was building a new music room that I wanted to share with my family. I wanted an amp that would not heat up the room so that the others in my family would hang out with me. I previously had an OTL amp which sounded excellent but turned my room into a sauna. To make a long story short, I have no regrets. I am now using SET amps (Coincident Frankensteins). I was fortunate to find speakers later on that work very well with the Frankensteins. My wife will sit with me and listen to music now. In fact, she has been regularly asking to listen to music together. It's been a plus for our relationship.

Erik: People who do surveys for a living know how difficult they are because the way you ask the question often shapes the answer given. First, I think it is unlikely that a significant number of people start over from scratch when putting together audio systems so, the either/ or question may not apply. More likely, they build on what they have - upgrading what they perceive to be the weakest link. I don’t upgrade often but I have cycled between amps and speaker upgrades an equal number of times. I upgraded my speakers this year. Before I did that, I sent an email to the amplifier manufacturer asking whether my old amp could drive a 4Ohm load. They said yes but if they had said no, I would have restricted my search to 8Ohm speakers.

Imaging? Relaxed presentation? Bass juiciness? Best presentation of Nina Simone you ever heard?

Yes, yes, yes and who??

When I have the right amp in the system, its a lot easier to convince my girlfriend to spend the evening spinning LPs...