I am not sure regarding the kettle idea...
What I did notice (as a student) is that, when you use the tongue as advocated by Arban's and every other material of musical authority that advocates the use of tongue, it changes the minds focus off the lips.
The whole problem I see (at the start of my learning) is that I was too focused on making notes with my lips. Everything was air/lip based. So, when I wanted to go higher, brain said "Tighten the lip". Want to go higher again "tighten the lip". Want to go more high, "lips". Everything I did was manipulate the sound via the lips.
That method is good to make you a bad player or at least not a very developed one. It is limited.
My teacher I can say is one of the best teachers around. He started teaching me to use my tongue, to get it mobile, to have it moving in the mouth not just "focus on lips" as I have been. He is the one that told me the first aparatus is the Lungs, if we have bad air, no support etc, everything else will suck basically.
So get that right first, the fundamentals - Air. Allen Vizzutti and many others that he has had the privelege to talk with etc while doing his masters in teaching and everything else, always focus on "air". Air is like 85-90% of everything if you wish to have a big, round, solid sound.
The second aparatus he told me is the tongue. - The tongue is like traffic lights on a street, it directs traffic (air). So if you want to play higher, direct the air higher and so forth.
The third aparatus is the lips. - The lips - embouchure - however, is the last bit of information before the air leaves the mouth. Once its out of the mouth that is the end of controlling it.
So he has been working on me to think from 2nd line G - to use "oo". When you go down from there (like the C below staff), you go "aw". This places the tongue in slightly different positions. Below C we just drop the jaw.
Embouchure should have the sides set, so we do not collapse the pillers (kind of like a bridge, if hte pillars collapse everything will be a mess). I used to collapse mine to go lower. With the work and teaching method, I now understand why "Drop Jaw". It allows the embouchure to remaing in a healthy position while opening the apeture up to get lower notes. It also keeps embouchure relaxed from being over-used. Bigger hole, better sound.
Now as we go from 3rd space C and above, the idea is "ee". This raises the tongue in the mouth thus changing pitch.
The reason I can not say its like the kettle, is, because, by default, I do notice even by using the tongue in different positions, that the lips do change as well. So they do not stay identical as it would in a kettle.
The idea is however that, if you think OOO, Awww, Eeee etc, (the manipulation of the tongue, the thing that controlls the traffic), the least stress will be placed on the lips. So yes, the lips change accordingly, but, because you are not thinking about the "lips" (as I used to), you are basically allowing the lips to do 'minimal' work because you are thinking tongue.
This method has allowed me to develop much further than where I was, my sound is a lot richer, my lips do not die as early as they used to, G above the staff now sounds full in tone and rich, rather than as though its being closed off (by the lips being manipulated too much).
So I do not say the lips do not move. I say as my teacher has shown and I have experienced, by thinking "tongue controls the traffic", less stress will be placed on the lips. Which means, the Lips will do what they need to do automatically, they will change without you thinking about it as you do the controlling with the tongue. Thus you control the sound at the 2nd aparatus, not at the 3rd and final one where the air has left your mouth and all too late to think about.
By using the tongue the lips basically do what they need to by default (so change the tongue, lips will move on their own accord to where the tongue is wanting to go). However, while we think about "control it with the lips", they get over-used, poor sound quality gets made and really, by that stage, the air is out of the mouth already, which is too late.
I am sorry if I have doubled up on stuff... I find it hard to try to explain in written form. And I still don't know if even this made sense.
From my own experience, any good teacher will explain, and think tongue to the student. It has transformed my playing from the days when I used to think "lips".
Like my teacher said and I have experienced, not everyone that is a professional principle orchestra player knows how to teach.
These are all methods of teaching to develop a players best ability and tonal production. If one can not or does not care to address the tongue (or any other factor that is part of playing trumpet, like the A frame, lungs, intercostal muscles etc) and how it works, why it works, how it is all connected and so forth... look for a new teacher, they are no good as they can not explain to the student what is 'actually' going on leaving the student with a ? in his/her mind because the teacher could not explain it or make them aware of it to consider. Sometimes I find, professionals just take everything as a given, it all works, they do not have to think about it. That is all good and fair-enough, but that stuff is still happening (just subconsciously to the professional). The student is not a professional though and is learning how it all works, so they need to be aware of everything before it can become "natural" to them, and if the teacher is not making them aware of it... what good is the teacher?
Bottom line is, everything must work in sync. #1 - Lungs, #2, Tongue (what happens inside the face BEFORE the air leaves the mouth and produces tone) #3 - Lips. Think about the first 2 being correct and the lips will by default be right and not over-used. The thing you want to do the least work with is the lips. If you can do everything via lungs/tongue, you will be able to play much longer at a comfortable, less stress rate.
Air starts in lungs, goes past the tongue then through the lips. Thus importance should be Lungs #1, Tongue #2, Lips #3 in order of how the air goes through your own body.