With natural splines, is the reason why we don't have locality because we have constraints 2 and 3 (C1 and C2 continuity)? This means that when changing one control point, both polynomial functions adjacent to the control point change, rather than only one of the polynomial functions.

It is because the constraints of tangent/curvature i and tangent/curvature i+1 are equal to each other. Since coefficients/values of different control points depend on one another, it is a global linear system and doesn't have locality.