The vowel formation wherein the mouth is wide open regardless of the pitch sung can be traced to an attempt to control the voice through overt direct mechanical manipulations.
We try hard to figure out how to sing, and then get stuck in a certain way of doing it which is not healthy, but is all we can figure out at the time. I often think of “popular myths” like the three finger rule where the mouth is supposed to open that wide.
There is more mouth opening going up the scale, and sometimes people need to be reminded to open their mouths. But, if this becomes some kind of magic trick to try to fix maladjustment at the sound source, then the “3 fingers” has overstayed its welcome. Just by opening our mouths wide will not fix the registration. And, if we were to try to sing with three fingers in our mouths, then we would look silly and cause rigidity of parts due to hyperextension.
So, yes, our mouth is sometimes pretty open, and sometimes not so much. The mouth opening is determined by the registration and vowel adjustment for the given pitch. The mouth opening is not something we do but becomes more of a subconscious reflex activity that happens as a result.
We notice what happens after a good experience and remember that feel and thought form in our voice. We begin to trust that our body knows now what to do, and we can mostly focus on expressing the music and text of a song. Our mouth will open as much as is necessary.