This could actually be quite important, or at least another variable.
Like most things in audio, sound stage is more tricky a topic than one might think. As long as we are just talking left/right localisation things are fairly straight. There are mainly (but not exclusively) two mechanisms involved, interaural level difference (most relevant above 1 kHz) and interaural time difference (most relevant below 1 kHz).
When we audiophiles talk about "sound stage" we don't just think left/centre/right, though. We are sure that we can hear the depth of the sound stage in various layers, with instruments in front of and behind the speakers. We even are convinced that only with the
right DAC/Amp/cable riser/speaker placement the height of the sound stage will just fall in place with
good recordings. And of course, this isn't entirely wrong.
It's just that for sound directly in front of us or behind us, above us or below us (should that happen) and everywhere in between
always has the
same level and
zero time difference in the left and right channel. How can we hear a difference in sound stage height if both speakers are playing exactly the same signal? Stereo is of no help here! Same for the depth of the sound stage. If one instruments sounds further away from you than the other but both are directly in front of you, this has nothing to do with stereo at all.
So, what's happening then? Our hearing is evaluating additional clues. Most prominently, the shape of our auricles and the shape of our head come into play. Our hearing does in fact have a different "frequency response" (or more formally: transfer function) for sound coming from different directions. This is most important for front/back and above/below localisation of sound events, but it also plays a (somewhat subordinate) role for left/right localisation (think: sound stage extension ...).
For spacial hearing our brain tries to gather as many clues as possible, filter out conflicting information and create the most likely hearing experience for us. This is not trouble free. The more contradicting the different clues are, the more exhausting the listening experience can be. Regarding the depth of the sound stage, clues are even more vague. Sound sources further away from us are usually lower in level and higher frequencies are tamed more than lower frequencies. The next time you are in a real concert hall, close your eyes and try to find out how good our hearing
really is at telling the distance to a sound source. Spoiler: Not very good at all ...
What all these additional mechanisms have in common is that they all rely on things that we
had to learn! It's not pure acoustics leading the way. If you never heard a trumpet in your whole life, how would you know how different it sounds played in front of you or behind you? Of course, we are learning from experience and can extrapolate from all the sound events we have heard before. There are other even more obscure things our brain does to help us perceive an unambiguous illusion of sound around us. It's also a matter of concentration. I even heard that newer studies found a new additional influence on the so called cocktail party effect (in a group of many speakers we can filter out everything but the one we want to listen to). It appears that our brains even takes into account the direction we are looking. And I have left out may other factors like the influence of reverb or echos.
Why this long story, then?
Localisation of sound events is not just about phase and level. It's not even just about stereo. Many things come into play, it's a real mess. Even slight variations in certain frequency ranges can change our perception of spacial hearing, masking effects can have a positive or negative influence. Often we just need to take away few factors irritating our brains like certain characteristics of the listening room (while taking away reflections completely is not a good idea). Room correction and subwoofers can be really helpful here, but nobody said this is easy. You didn't like the outcome of your previous RC run, but there are lots of things you could change and try again (target curve, frequency range, max. gain and max. Q for correction). The most pleasing crossover frequency, delay and level for your subwoofer (to be detirminded before running RC) can all have an influence. An influence that is much, much more important than the difference between any two decent amplifiers.
Remember, it's all just really an illusion (which is not a bad thing).
Edit, for reference and further reading:
en.wikipedia.org