Gain / Amplification

If it is just divider at the input then first point doesn’t apply, but reduction of IC noise pickup is still important

It totally applies there @kijanki - imagine 10 nV on the IC and then amplifying that small, medium or large.

Small is always smaller than large, and noise will be smaller.
The only reason to go to medium or large is if it not loud enough.

That is a good amp you have, but maybe you can see (hear) if it makes a difference with the music not playing?
(Turn the pre amp +way+ up.)

It totally applies there @kijanki - imagine 10 nV on the IC and then amplifying that small, medium or large.
 

Then you're talking about noise pickup by interconnect.  That was mentioned as second reason.   First reason was that amp with fixed gain and signal divider at the input will react the same way to electrical noise pickup inside of the amplifier at all positions of the input divider, while amplifier with lower overall gain will be less susceptible to electrical noise pickup inside of the box.

It’s about matching the volume of various inputs so you do not need to adjust the master volume much, certainly avoid any loud jump from an input change

Often you are talking about more or less attenuation rather than gain.

Much equipment, like my McIntosh Preamps SS C28 and Tube mx110z have ’trim’ controls for the phono and aux inputs to match them to each other. They were located near the front on the top, you pull the unit forward a few inches, adjust them and push the unit all the way back.

My Tandberg Receiver had trim controls in the back.

Any unit I have had, like Onkyo DX-7500 CD Player

https://www.hifi-classic.net/review/onkyo-dx-7500-430.html

had 3 outputs, rca; direct; and variable. I couldn’t hear any difference, neither could my friends.

Likewise, I had 3 oppos, and the output variations sounded the same to me. Turning off the display may be measureable on the bench i.e. .00002, but its marketing IMO.

Gain usually comes with additional noise. The less gain you have overall the less noise.

Phono stages have the most gain, and most prone to noise, followed by line level preamplifier circuits.

That is all in theory though. Over the last 20-40 years solid state has gotten very very quiet so it’s not usually worth sweating these details vs. say room treatments, listener location, etc.

The one pet peeve I have though is that most preamps still have WAY too much gain, maybe due to the days when radio might pick up a weak station and you had to compensate for it with the volume control.  If I used a tube pre especially I'd be tempted to reduce the gain quite a bit.

is there likely a quality-of-sound difference between utilizing maximum gain (say, out of a CD player, to keep it a simple scenario) and needing much less amplification from your amp, or is it in any way different to set your CD gain/output volume at a lower level and let your amp do its job?   Let’s say balanced connections, so that’s added gain as well.    

If the balanced connection supports AES48, the balanced line standard, there may be no difference of signal level between balanced and single-ended.

The newer more sophisticated digital volume control systems are a bit better at resolution. Older ones cause a loss of resolution as you turn them down since bits are subtracted.

This is why there is a market of line stage preamps since they often have better volume controls as well as often are able to minimize cable interactions better than most sources.