Your teacher is correct.
But I am not worried about your teacher, or even your high notes. I am worried about your attitude.

I have told you this before but it fell on deaf ears, but I'm going to tell you it again:

Singing is way more than "hitting some high notes". Your problem isn't range. Your problem is that you haven't learned how to just SING in your comfortable range and CARRY A TUNE. The stuff you are obsessed with is only 5% of the entire thing. a singer isn't better because they can hit a G#5 instead of another singer who "only" sings a G5..

This doesn't mean stop working on extending range, keep going by all means - but your main focus needs to be SINGING WELL.

How often do you listen to a song within your comfortable range and hit pause and hit record and try to sing it back and hold up to the original standard? Then play back your recording and listen to where you were off and where you were on? How close can you get it? Listening to the inflections, the fades, swells, intensity of the notes, breathy or clear? Have you created a set list of songs to work through everyday and can make sure you can sing through them and pace yourself through the entire set? Can you sing it up to a good standard?

I'm sure this post will piss you off but it needs to be said because too many vocal coaches know exactly what I'm talking about but will say nothing and wll proceed to just teach you technique because they feel it's out of their place to speak up. I used to be like that too but then I realized I didn't want students who could only sing scales really high but couldn't sing simple tunes

I recommend you start a project to do some covers of songs that are well within your range and to see how you can handle them and post them up, I can give you feedback. A good thing to think about would be: if you had to perform a concert by next week and the highest notes go up to A4 would you be able to deliver the songs up to a high standard? If the answer is yes I would love to be proven wrong - I want you to succeed. if the answer is no then I think you'll see my point in this post.

It's better to think of it not as a break but as gear shifts or areas where resonance needs to shift. So you have that one at F4, then at around A4 there's another, then around C5 there's another then F5 another etc. They change slightly depending on how much weight you roll up. As you get better rolling weight up you feel like there's less of these gear shifts I guess because they just happen so smoothly like in a car you don't notice a good driver changing gear.

At first it can feel like you are literally "stretching chest" because most people would typically go into falsetto at around their first "break" being F#4. So when they finally build the coordination to thin out the voice properly and take it up it feels like you are stretching something (in a good way not in a strainy yelly way) this stretchy feeling when done properly is a good indicator that you are using compression instead of letting the voice become "floppy" and just breaking. It should stretch like a rubber band but never feel tight in a bad way.

believe it or not singing can get to the point where it does feel relaxed. never effortless but it is a very reasonable amount of effort nothing that should feel ridiculous.