Sorry but you're missing the forest for the trees here.
The locality of a bezier, even with symmetric constraints, amounts to the fact that the user's specified positions (the endpoints) and tangent directions are preserved exactly and can be modified independently. This makes detailed edits easy. And because curvature beziers only differ from classic beziers in the tangent lengths (not directions), the same properties hold.
The "7 control point span" you talk about is really a 5 point span, because adjusting the lengths of the bezier tangents at the far ends serves to explicitly preserve the user's intent, namely the radius of curvature.
Catmull-Rom e.g. ties together the tangent at a control point with the position of the previous and next control point. And B-splines don't guarantee going through any control point at all except the endpoints, with the tangents not even independently controllable.
The locality of a bezier, even with symmetric constraints, amounts to the fact that the user's specified positions (the endpoints) and tangent directions are preserved exactly and can be modified independently. This makes detailed edits easy. And because curvature beziers only differ from classic beziers in the tangent lengths (not directions), the same properties hold.
The "7 control point span" you talk about is really a 5 point span, because adjusting the lengths of the bezier tangents at the far ends serves to explicitly preserve the user's intent, namely the radius of curvature.
Catmull-Rom e.g. ties together the tangent at a control point with the position of the previous and next control point. And B-splines don't guarantee going through any control point at all except the endpoints, with the tangents not even independently controllable.
This is why artists like beziers.