It should work to drag the off-curve nodes so that they touch where the corner would be if the rounded rect was a square/rectangle.

Using regular beziers you should be able to get at least G1. Symmetric smooth cubic bezier nodes should be C1. With regards to G2 Inskcape and Coreldraw have b-splines, not an Illustrator user from what I could find seems like Illustrator lacks it.

Blender has tools for at least basic nurbs modeling.

I am slightly skeptical on the need of this in 2d vector drawings (outside few very specific use cases). Big reason for having higher degree continuity of 3d surfaces in industrial design is that looking at reflection in mirror surfaces (like cars) makes the difference very obvious. For 2d drawings you can't really look at the side of it and see 1d reflection.