Would you be aware of any rationale for why that might be? I was under the assumption that averages are defined regardless of the number of dimensions.

I would rather invert the question, why would there be a broadly used definition unless it’s useful?

Averages can also be defined in different ways, with medians, logarithmic means, squared means, Root Mean Square , and most of statistics, beyond the simple arithmetic one.

From an intuitive standpoint, how would you even go about to describe an n-dimensional curve as something useful in one dimension (scalar).

What meaningful, direction independent, quantity would you describe to differentiate between a cylinder and a sphere, or other shapes?

And I mean these questions not to be disparaging but as a guide, find a useful quantity and a meaningful formulation and you have created a mathematical tool. Get it to do cool and/or useful things, and people will start using it. As enough people adopt it, it becomes the widely used definition that we can point to the next time someone asks this question.

A simple way is to solve a practical problem with it, a more mathematical way is to express a neat relation, simplify a clumsy proof and/or solve one of the unsolved problems.

Normals are useful as they give the orientation of out (and are also useful in creating subspaces), so when working on finding the outside of surfaces, which we’ve spent many decades on, it came out as a commonly used tool. They’re now also used in Machine learning theory to describe Solution Spaces to Parameter spaces, although I don’t know enough to know if it’s actually useful or just trendy.

Which formulation are you referring to?

The summary formulation: ∇² = f’‘(x) + f’‘(y) + f’'(z).