If I want to be able to send the signal from a CD player (or record
deck) to

a) amp (for "pure" listening)

- OR -

b) rough old DJ mixer (for occasional parties)

.... is it an acceptable option to use a "Y" splitter cable out of the
CD player (record player)?

I want to avoid doing any replugging when moving from "audiophile
mode" to "party mode". When in "audiophile mode" the mixer would be
turned off. I have tried to look around for an answer to this, and
while people seem to react against Y splitters, I haven't been told
exactly WHY they are such a bad idea.

Is there a good electronics engineering reason to avoid them? Would it
make any detectable difference in the real world (double-blind testing
etc etc)?