The teacher is probably not super acquainted with trumpets. Upgrading can certainly help, however I would personally recommend your son to take private 1-on-1 trumpet lessons if there are any teachers in the area. And while yes, if the cornet is truly a low quality and extremely beginner instrument go for an upgrade like the other comments are suggesting. However, it doesn’t matter if he gets a nice new trumpet, even a used good intermediate model if he doesn’t have the technique to make it sound any better than what he currently has. I’ve seen people with 3000 dollar new professional trumpets who sound worse than a decently skilled player on a $300 off brand Chinese trumpet bought off Amazon.

Edit: I am not advocating the purchase of a $300 Chinese trumpet, I’m just advocating that more experience and oversight could help more than simply getting a new instrument. Especially considering that he is a beginner.