It is well known that there are many interpretations of quantum mechanics. I'm wondering if there is a specific reason why the Copenhagen interpretation is the most popular. Why is it that the wavefunction interpreted as a probability distribution is such a useful description. This question came to me when I was trying to write a paper and got a strange wavefunction (it wasn't normalizable, had singularities etc.). I'm a mathematical physics grad student so I was usually more concerned with mathematical elegance than physical interpretations, but this question has been bothering me.
I'm not sure its more complicated than the fact that the Copenhagen Interpretation is the oldest and most widely taught.
Couple this with the fact that many physicists don't spend too much time worrying about things like Quantum Interpretation, and you're left with a population that, when pressed, say they follow the Copenhagen one.
This is not to say this is the way it should be, but it is the way it is.
But, I am a little troubled by your statement:
Interpreting the wavefunction as a probability density is in no way unique to the Copenhagen interpretation. The disagreement between different interpretations usually center on questions like collapse, or perhaps more subtly, what, precisely the square of the wave function gives you the probability of. Whether it is local to the observer, or a statement about possible futures, or worlds, or the action of some stochastic potential or what have you. But I can't think of a single interpretation what wouldn't interpret the square of the wavefunction as a probability of something.