There are two separate questions here. One is how much moving control points creates expected changes in the same direction. Béziers nail this, as the position of a point at t is a linear combination of the control points, with the Bernstein polynomials as weighting functions. So it always feels like direct control. With my mapping, you get this for a nice big chunk of the parameter range – small to moderate angles and control point distances. But this property does fall apart when pushing to extremes.

The other question is whether the control is local. As others have remarked, this has much more to do with the way the curve is embedded into a spline than the curve family itself. In particular, it is deeply affected by the continuity constraints. As payment for the local support, cubic Béziers only give G1 continuity. Euler spiral splines, by contrast, are G2 but changes do cause those ripples.

The pen tool prototype linked in the blog post suggests giving designers more choices. If you specify all control points, you get G1 just like a cubic Bézier. But it also gives the choice of specifying one control point on a smooth endpoint, and solving the other for G2 continuity. In my experience, it feels more like local control than Euler spiral-like splines. When you want smoother curves, you do that, and when you're willing to sacrifice continuity for local control, that's also possible.