I like to think of it as dot product being the true "natural" space to compare magnitude metrics, whereas absolute value is just a human construct conceived for our mental convenience. A smooth parabolic bowl vs an unnatural sharp conical tip. Also shows up in standard deviation etc.

Aside: I wonder if complex values neural networks with activation function just being sum(inputs)*conj(sum(inputs)) with threshold normalized by sqrt(num_inputs) could be the most universal, where incoherent inputs will average an absolute value of sqrt(N) and coherent inputs are N like lasers? (square amplitude would be N vs N^2 between uncorrected and correlated population)

why not raise to any other even power ?

That's mnemonic not intuition.

That doesn’t answer the title question of why it’s quadratic wrt speed.

> Why not take the absolute value? Nature hates those

And yet inverse distance laws for potential energy for gravity and electric fields use the absolute value because they require an unsigned distance and how you treat the singularity at zero is extremely important to the structure of those interactions