I've read just about this entire thread and would like to give my experience.
I can play to F over high C. I don't know that I own the F. Most of the time it's there.
I play lead in a big band. We have charts that go higher than my F. I wish I could hit the notes. Here's the question, is it required? I have the job and yet, I can't hit the notes. When it comes up, I take it down an octave. If it messes with the harmony, I have to talk to the section about lowering the other parts. I don't know if the other sections know I'm doing it. I really don't think they care.
I also play a ton of classical jobs. If I focused on the double C, what would happen to my sound? Maybe nothing, maybe something. I was in a room with Patrick Hession while he was messing around. He played up high and low and all with a good sound. I think he's the odd one but, I guess it can be done. He was sitting in a chair with almost no effort up to double C. I don't think he was using a tight stomach or tons of air. He just played like we would around the staff.
When I switched to Monette equipment I don't think I got more notes. The notes became more consistent and came out a little easier and, more "in tune".
As fare as a shallow mouthpiece, I could never play all night on my regular mouthpiece. I can get the F but, it wouldn't last. On the shallow mouthpiece, I can last forever and cut through the band. It's actually more of a V cup than shallow. I think the most important thing is that it fits.
Now that you've read my random ramble, if you come up with a way to hit the double C, please let me in on the trick.