This is very similar to the "Minimum Energy Curve" fairness metric, which minimizes sum of of bending energy, which in turn is proportional to curvature squared. And, thanks to basic physics, curvature is proportional to the centripetal force given a constant velocity.

The problem with MEC is that it tends to optimize for low tension; for inputs that are not well behaved, it sometimes optimizes for infinitely low tension, ie an infinitely long road. So there are other metrics, like Minimum Variation of Curvature, that work better for actual spline use, even though they have an energy metric that's "worse."

These curves are absolutely useful for path planning. A lot of the literature on "clothoids" from a hundred years or so ago is on transition curves for railroads. The G2 continuity condition basically means that you never have to suddenly turn the steering wheel to follow the curve, it's always moving continuously.

For what it’s worth it makes beautiful sense to me. Both as a motorcyclist and also as someone who has thought a lot about splines.

My favorite spline editor is the Bezigon tool in Macromedia Freehand of yore. It sprang to mind as I read the passages you wrote.