That may technically be correct, but it is incorrect in the real world. I submit that error is error in the real world. Mathematics can go jump off a cliff unless it wants to be helpful. :)

Zero average error conveys something important though: the error that there is, isn't biased positive or negative.

That's language failing us, not maths :-)

What are you talking about? Error is a metric.

By this logic, a broken 24 hour clock stuck at mid day has 0 error.

The average of its errors is 0- ie it is not biased. Ofc, the average absolute error, which in English one could very reasonably refer to as "average error" is much greater than 0.