Let me expand on playing musically in the upper register. I've known players who developed range by arpeggios only. It works to a point. They worked on it as weight training. The problem was that they increased the stiffness in their lips to a point where they lost flexibility. This approach also easily leads to requiring an embouchure shift. Think about it as you play your 1 octave arpeggio or even scale the starting note gets higher and higher. You take a breath and play the next series. You take another breath .... There is an almost overwhelming desire to make subtle changes on each breath. Here is a test start on high c and play an arpeggio up then play it down to low c. Was it slow to respond or of a poor tone quality? Then you are playing with an embouchure shift.
There are some ways to avoid this. Always set your chops for a g on top of the staff. It is only an octave to high g and only an octave and a half to low c. When you do practice arpeggios or scales up for range always play them back down to low c or below on the same breath. This will help you to learn to play all registers with one embouchure. The reason to play simple songs one or two octaves up is to learn to play musically. Even if it is Mary had a little lamb there IS phrasing. Play old Jr. High or HS solos an octave up. Tired of your Concone studies play them up an octave. That's a real workout. Work on tonguing drills in the upper register. A range to 2 octaves over high c is no good if you can't tongue well.
Work on your mindset. Has this ever happened to you? Look at the last note in this chart . Then you worry the entire chart about 1 lousy note. Right up to the time when you have to play it you consider weither or not to play it down an octave. Then you take a deep breath and make a real ugly sound. You just committed psychological paralysis.