Which component is most responsible?

Yes, you need a good choral recording recorded a certain way to even have a chance, so make sure that is in place first, otherwise you will spin your wheels.

Here's a good recording to try off the cuff:

Mitch Miller

Here's another:

MTC

Power cord.

(and/or wall socket).

Yes, reasonably clean power however achieved is a must for this as well.

Following up on Brownsfan's suggestion I've picked out a couple of acapella choral recordings you might like (I do anyway) which are probably still in print and might serve your purposes, although I must admit, as I suggested in my first post, that if I were trying to pin down specificity I probably would not use choral music for one simple reason. Choral music tends to be a powerful blend of voices and single voices rarely stand out especially with minimal mic'ing techniques, but even with spot mic'ing, except when used for the soloists. Anyway, here goes....

Rachmaninov's All-Night Vigil, by the Estonian Phil Chamber Choir conducted by Paul Hiller on Harmonia Mundi.

Grechaninov's Passion Week by the Pheonix Bach Choir & KansasCity Chorale conducted by Charles Bruffy on Chandos (SACD)

Sibelius Choral music by YL Male Voice Choir conducted by Matti Hyokki on BIS. This disc has both acapella as well as choral accompanied by orchestra Lahti SO conducted by Vanska. I'm partial to the acapella performances, especially of Finlandia-Hymni a transcription for all male choir of the original, the orchestrated Finlandia. I understand the transcription was to be used by soldiers matching to the war with the Soviet Union in WWII. I still get choked up when I listen to it.

"Which component(s) is most responsible for hearing clear, distinct separation of voices in a chorus or when listening to multiple background vocals, as I often hear audiophile speak of."

All of them.