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.

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.