Competition – as any other manifestation in the universe is neither evil nor good. It is neutral. It is just us that either like or dislike it.
Again, as usually is the situation in the Universe, our liking or disliking of its manifestations has little to do with it actual occurrence.
Competition will happen (like it or not) whenever a resource is in limited supply in relation to its desirability.
I don’t think music by itself leads to competition, although its peripherical activities always do.
If there is limited supply of students, piano teachers will compete either openly or covertly, either knowingly or unknowingly (and sometimes in deep denial).
If there is a limited supply of public, performers again will compete, and try to have an edge over their fellow performers.
However, students of music have little to compete about music itself (unless teachers are in limited supply, or there is not enough printed music to satisfy the market).
What should a student compete about? Who plays best? Who pays in time (hey! there is an idea!

) We know from competitions that this simply does not work because there is no way to specify in a non-subjective manner what “best” means.
It seems to me that although some people do thrive on competing and that could be used as some sort of incentive for this sort of people, it is not an attitude that I personally would care to associate with the teaching/learning of music.
In fact, no one thrives on competing. What people actually thrives on is
winning. As someone once said to define sportsmanship: “It doesn’t really matter if you win or loose as long as you win”.

My first reaction (as you may have guessed), is : “bad idea”.

However, maybe I have not understood your idea (you didn’t give much detail).
Are we talking about the same thing?

Best wishes,
Bernhard.