why would there be a broadly used definition unless it’s useful?

Knowledge for knowledge's sake? I find generalizations to be very satisfying.

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.

For the sake of this context, both me and the video are referring to the simple arithmetic mean.

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

Hrm, well, it would entirely depend on context.

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

I don't follow this point. Would it not be entirely context dependent?

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.

Imo, this is approaching it from, say, a physicist's, or engineer's perspective, rather than from a mathematicians perspective. Math doesn't need to have direct applied use for it to be interesting.