Regrettably, “intended purpose” is highly subjective.

Sure, << for stream output is pretty unintuitive and silly. But what about pipes for function chaining/composition (many languages overload thus), or overriding call to do e.g. HTML element wrapping, or overriding * for matrices multiplied by simple ints/vectors?

Reasonable minds can and do differ about where the line is in many of those cases. And because of that variability of interpretation, we get extremely hard to understand code. As much as I have seen value in overloading at times, I’m forced to agree that it should probably not exist entirely.

Depends on what you're trying to understand.

Let's say I have matrices, and I've overloaded * for multiplying a matrix by a matrix, and a matrix by a vector, and a matrix by a number. And now I write

  a = b * c;

  a = mat_mult(b,c);

But if I take the exact same line and try to understand exactly which functions get called, because I'm worried about numerical stability or performance or something, then the first approach hides the details and the second one is easier to understand.

This is always the way it goes with abstraction. Abstraction hides the details, so we can think at a higher level. And that's good, when you're trying to think at the higher level. When you're not, then abstraction just hides what you're really trying to understand.

> we get extremely hard to understand code

The thing is code without operator overloading is also hard to understand because you might have this math thing (BigIntegers, Matrices) and you can't use standard notation.