why is 63-37 eutectic solder not universally preferred?

Because sometimes you want mush.

Eutectic is either solid, or liquid.

Formulations off eutectic have a range of mushiness, which can be of use.

....
Not wanting to be confrontational, just curious. Wikipedia lists melting
points as:

63/37: melts between 180-185°C
60/40: melts between 183-190°C
50/50: melts between 185-215°C

I'm not aware of any soldering process that is able to control
temperature so closely as to not melt one, while melting another of
those formulations. Can you elaborate?

It's not whether you can melt it, it's how it hardens. Eutectic solder
hardens very suddenly. 60/40 solder goes through a fleeting stage of being
viscous but not yet completely hard. I'd like to hear more about this, but
I'm told some people prefer the "feel" of one vs. the other.