How to draw a freehand curve fitting the surface of the model?

See the enclosed part for reference on how to create your curve on surface.

The tool you are attempting to use is restricted to a domain of 1 unique surface. The issue is due to how surfaces are handled in the geometry kernel. Even though you have created a quilt by copying the solid surfaces, each surface patch has a unique internal feature id. When you are attempting to create a curve on surface only surfaces can be selected, not quilts.

You can create curves on surface and connect them across the surfaces to maintain continuity. If you use multiple on surface curves and connect them it will work.

The curve on surface within style is by design tailored to work with surfaces created in style. Style features are super features that behave differently than core Creo features.  While they work with core Creo features they were integrated into Creo from a separate software product that was created as a surface modeler. This is why I used a style drop curve in my previous example. The curve used to drop can be planar or a 3D curve.